FUNCTIONS OF A COMPLEX VAKIABLE MACMILLAN AND CO., LIMITED LONDON BOMBAY CALCUTTA MADRAS MELBOURNE THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO DALLAS SAN FRANCISCO THE MACMILLAN CO. OF CANADA, LTD. TORONTO FUNCTIONS OF A COMPLEX VARIABLE BY THOMAS M. MACROBERT M.A., B.Sc. , LECTURER IN MATHEMATICS IN THE UNIVERSITY OF GLASGOW FORMERLY SCHOLAR OF TRINITY COLLEGE, CAMBRIDGE MACMILLAN AND CO., LIMITED ST. MARTIN'S STREET, LONDON 1917 COPYRIGHT GLASGOW: PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE AND CO. LTD. PEEFACE THIS book is designed for students who, having acquired a good working knowledge of the calculus, desire to become acquainted with the theory of functions of a complex variable, and with the principal applications of that theory. In order to avoid making the subject too difficult for beginners, I have abstained from the use of strictly arithmetical methods, and have, while endeavour- ing to make the proofs sufficiently rigorous, based them mainly on geometrical conceptions. The first two chapters are intended to familiarise the student with the geometrical representation of complex numbers and of the simpler rational and irrational functions of a complex variable. In Chapter III. the properties of holomorphic functions are established; these properties are then used to define the Exponential, Circular, Logarithmic, and other transcendental func- tions for the domain of the complex variable, their properties as functions of a real variable being assumed to be known. It is thus possible in Chapter IV. to make use of these functions in examples on integration ; such examples are both interesting and important, and it seems desirable to introduce them to the student in a manner that does not involve the difficulties of complex series. As a preliminary to Green's Theorem I have given a short account of curvilinear integrals. Two proofs of Cauchy's Theorem are given, only the first of which depends on Green's Theorem. A large number of examples on contour integration are worked out, and here, as throughout the book, the text is plentifully illustrated by diagrams. vi PREFACE In view of the very full exposition of the subject given by Dr. Bromwich, it has been thought unnecessary to give a de- tailed account of infinite series. A summary of those theorems which are used in the book will be found at the beginning of Chapter V. ; the theory of uniform convergence is dealt with in Chapter VI. The remaining chapters are devoted to the applications of the subject. Chapter VII. includes, among other matters, the theory of Analytical Continuation ; various examples of the applications of that theory are given there and in Chapters VIII. and XV. The asymptotic expansions of the Gamma Function in Chapter VIII. and of the Bessel Functions in Chapter XV. are worked out for complex values of the variable. Chapters IX. to XI. deal with Elliptic Integrals and Functions. In Chapter IX. the student is shown how to reduce and evaluate elliptic integrals. In Chapter XI. I have established the exist- ence of the Jacobian Functions by considering the values of the Weierstrassian Function when one period is real and the other is purely imaginary. The last four chapters of the book contain a discussion of the theory of linear differential equations. As the most important of these equations are of the second order, it has been thought unnecessary to consider equations of higher order than the second. The Hypergeometric Function and Spherical and Cylindrical Harmonics are discussed as they arise through the solution of their differential equations; other properties of these functions are given in examples, with, in most cases, hints as to the methods of solution. No attempt has been made to deal with the applica- tions of these functions to physics, but it is hoped that the applied mathematician will find in these pages ready access to the instruments which he requires. Numerous examples have been given throughout the book, and there is also a set of Miscellaneous Examples, arranged to correspond with the order of the text. The writing of the book was undertaken at the suggestion of Professor George A. Gibson, LL.D., to whom I have been PREFACE vii indebted for important criticisms at all stages of the work. I have also to thank my colleagues, Mr. Robert J. T. Bell, D.Sc., and Mr. Arthur S. Morrison, M.A., B.Sc., for their assistance in correcting the proofs. Acknowledgment has been made, in foot-notes to the text, of various sources from which I have derived assistance. Of the books which I have found helpful I would particularly name Lindelof s Calcul des Residus, Cauchy's Mtfmoire sur les inte- grates definies, Jordan's GOUTS d' Analyse, and Forsyth's Theory of Differential Equations. I have also made use of lectures by Mr. R. A. Herman, M.A., and Professor E. W. Hobson, Sc.D. In conclusion, I would express my thanks to Messrs. MacLehose for the excellence of their printing work. THOMAS M. MACROBERT. GLASGOW, September 1916. CONTENTS CHAPTER I. COMPLEX NUMBERS. SECT. PAGE 1. Definition of Complex Numbers 1 2. Geometrical Representation of Complex Numbers 1 3. Modulus and Amplitude. Vectors - 2 4. Geometrical Representation of Addition. Subtraction- - - 3 5. Multiplication and Division 4 6. Root Extraction 5 Examples I. - 6 CHAPTER II. FUNCTIONS OF A COMPLEX VARIABLE. 7. Uniform and Multiple- valued Functions. Path of Variation - 7 8. Transformations. Point at Infinity - ... 7 9. Geometrical Representation of Functions. Branch Points - 10 10. Roots of Equations - - - - - 16 Examples II. ----- 19 CHAPTER III. HOLOMORPHIC FUNCTIONS. 11. Limits - 22 12. Continuity . . 3 13. Uniform Continuity. Functions of Two Real Variables - - 24 14. Differentiation - - - - 26 15. Definition of Holomorphic Functions. Simply-connected Regions. Inverse Functions. Harmonic Functions - - 27 x CONTENTS SECT. PAGK 16. The Exponential Function - - - 32 17. Circular Functions. Hyperbolic Functions 33 18. The Logarithmic Function. The Inverse Tangent Function - 34 19. The Transformation w = Log z - 35 20. The Generalised Power 36 21. Conformal Representation 37 22. Singular Points. Poles. Meroinorphic Functions. Essential Singularities. Branch Points. Zeros. Theorems - 38 Examples III. 40 CHAPTER IV. INTEGRATION. 23. Convergence and Uniform Convergence of Sequences ... 42 24. Curvilinear Integrals. Differentiation under the Integral Sign - - 42 25. Green's Theorem - 45 26. Definite Integrals - 48 27. Cauchy's Integral Theorem. Theorems on Integration 51 28. Cauchy's Theorem : Alternative Proof - 54 29. Cauchy's Residue Theorem. Residue at Infinity - - 57 30. Evaluation of Definite Integrals. Theorems on Limiting Values of Integrals. Principal Value of an Integral - 59 31. Cauchy's Integral 67 32. Liouville'e Theorem - 68 33. The Fundamental Theorem of Algebra 69 34. Differentiation and Integration under the Integral Sign - 69 35. Derivatives of a Holomorphic Function 70 Examples IV. 71 CHAPTER V. CONVERGENCE OF SERIES : TAYLOR'S AND LAURENT'S SERIES. 36. Convergence of Series. Absolute Convergence. Multiplication of Series. Ratio Tests. The Hypergeometric Series 76 37. Convergence of a Double Series - - 78 38. Power Series. Circle of Convergence. Abel's Test. Multipli- cation of Power Series 80 CONTENTS xi SECT. PA GE 39. Taylor's Series - 82 40. Laurent's Series. Principal Part at a Pole 84 41. Fourier Series. Periodic Functions 86 42. Classification of Uniform Functions. Integral, Rational Integral, and Transcendental Integral Functions. Theorems 88 Examples V. - 89 CHAPTER VI. UNIFORMLY CONVERGENT SERIES : INFINITE PRODUCTS. 43. Uniformly Convergent Series. Continuity. Integration. Dif- ferentiation. Weierstrass's M Test 92 44. Uniform Convergence of Power Series. Undetermined Co- efficients 95 45. Additional Contour Integrals 97 46. Legendre Polynomials. Expression in Definite Integral Forms. Recurrence Formula 99 47. Expansion of cot z in a Series of Fractions 103 48. Mittag-Leffler's Theorem. Weierstrass's Zeta Function. Weier- strass's Elliptic Function - - - 105 49. Infinite Products. Unconditional Convergence - 107 50. Weierstrass's Theorem. The Gamma* Function. Weierstrass's Sigma Function ... 108 Examples VI. 109 CHAPTER VII. VARIOUS SUMMATIONS AND EXPANSIONS. 51. Expansions in Series by means of Residues. Theorems on Limiting Values of Integrals - - - - - - - 113 52. Summation of Series by means of Residues. Gauss's Sum - 116 53. Theorems on Roots of Equations - - - - - - 118 54. Lagrange's Expansion. Rodrigues' Formula. Integrals involv- ing Legendre Coefficients - - - - - - - 119 55. Analytical Continuation. Theorems. Recurrence Formula for Legendre Coefficients - 122 56. Abel's Theorem - - 125 Examples VII. 128 xii CONTENTS CHAPTER VIII. /-GAMMA FUNCTIONS. SECT. PAGE 57. The Bernoulli Numbers. Expression as Definite Integrals 132 58. Asymptotic Expansion of Euler's Constant .... 134 59. Convergent Integrals 136 60. Uniformly Convergent Integrals. Differentiation and Integration under the Integral Sign - - - - - - - . 137 61. The Gamma Function. Gauss's Definition. Formulae. The Function ^(z). Gauss's Function 11(2). Euler's Definition. Expression as a Contour Integral. Gauss's Theorem - 141 62. The Beta Function. Duplication Formula for the Gamma Function - 144 63. Asymptotic Expansion of the Gamma Function. Analytical Continuation of the Hypergeometric Function 146 Examples VIII. - 154 CHAPTER IX. INTEGRALS OF MEROMORPHIC AND MULTIFORM FUNCTIONS : ELLIPTIC INTEGRALS. 64. Integrals of Meromorphic Functions. The Logarithmic Function 160 65. Integrals of Multiform Functions. The Inverse Sine Function - 161 66. Legendre's First Normal Elliptic Integral. Inversion of the Integral 163 67. The Weierstrassian Elliptic Integral. Inversion of the Integral 167 68. Elliptic Integrals in General. Legendre's Normal Integrals 169 69. Complete Elliptic Integrals. Landen's Transformation 173 70. Legendre's Relation - 175 Examples IX. 176 CHAPTER X. WEIERSTRASSIAN ELLIPTIC FUNCTIONS. 71. Doubly- Periodic Functions. Theorem on Primitive Periods. Congruent Points. Period-Parallelograms 179 72. Elliptic Functions. Weierstrass's Elliptic Function. Theorems on Poles and Zeros of Elliptic Functions 180 CONTENTS xiii 73. Relation between #>(z) and p'(z). The Weierstrassian Elliptic Integral 183 74. The Addition Theorem for ^(2) - 185 75. Properties of Weierstrass's Zeta Function. Legendre's Relation. Expression of Elliptic Functions in Terms of Zeta Functions - 187 76. Properties of the Sigma Function. Expression of Elliptic Functions in Terms of Sigma Functions - 189 Examples X. . - 191 CHAPTER XI. JACOBIAN ELLIPTIC FUNCTIONS. 77. The Values of $>(w; oi lf o> 2 ) when Wj is Real and o> 2 is Purely Imaginary - - 194 78. Geometric Application - 196 79. The Jacobian Elliptic Functions sn u, en u, dn u. Periods. Poles 198 80. The Addition Theorems for sn u, en u, dn u - 202 81. Jacobi's Imaginary Transformation - - 205 Examples XI. 206 CHAPTER XII. LINEAR DIFFERENTIAL EQUATIONS. 82. Continuation of a Function by Successive Elements - - - 208 83. Homogeneous Linear Differential Equations of the Second Order. Existence of an Integral in the Domain of an Ordinary Point. Integrals at Infinity. Analytical Continuation of the Integral 209 84. Solution by Infinite Series, Legendre's Equation. Legendre's Function of the First Kind 213 85. Theorems on Fundamental Systems of Integrals - 215 Examples XII. - - 217 CHAPTER XIII. REGULAR INTEGRALS OF LINEAR DIFFERENTIAL EQUATIONS. 86. Integrals in the Neighbourhood of a Singularity. The Funda- mental Equation. The Associated Fundamental System 219 87. Regular Integrals. Conditions for Regular Integrals - 222 88. The Method of Frobenius. The Indicia! Equation. Solutions Free from Logarithms ........ 225 xiv CONTENTS SECT. PAOE 89. The Gaussian pifferential Equation. The Differential Equation of the Quarter Periods of the Jacobian Elliptic Functions 228 Examples XIII. - 232 CHAPTER XIV. LEGENDRE'S AND BESSEL'S EQUATIONS: EQUATIONS OF FUCHSIAN TYPE. 90. Legendre Functions of the First and Second Kinds. Recurrence Formulae .... . 234 91. Bessel Functions of the First and Second Kinds. Recurrence Formulae. The Function GJz). Relations between the Bessel Functions. The Zeros of J n (z) - 236 92. Equations of Fuchsian Type - 243 93. Riemann's P-f unction. The Twenty-four Integrals of the Hypergeometric Equation. Analytical Continuation of the Hypergeometric Function .... . 244 94. Spherical Harmonics. Legendre's Associated Functions of the First and Second Kinds - 249 Examples XIV. 252 CHAPTER XV. SOLUTION OF DIFFERENTIAL EQUATIONS BY DEFINITE INTEGRALS. 95. First Method of Solution. The Branch Points of the Integral - 255 96. Gauss's Equation. Definite Integral Form of the Hypergeo- metric Function - - 258 97. Legendre's Associated Equation. Expressions for P n m (z) and Q n m (z) .--- 259 98. Second Method of Solution 266 99. Bessel's Equation. Definite Integral Expressions for J w () and G n ( Z ) --- 266 100. Asymptotic Expansions of G n (z) and J M () 271 Examples XV. 275 MISCELLANEOUS EXAMPLES - " - 277 INDEX - 294 CHAPTEE I. COMPLEX NUMBERS. 1. Definition of Complex Numbers. A number of the form p + iq, where p and q are reai.and i is a root of the equation i z -fl=0, is called a Complex Number. If g = the number is said to be purely real, and if > = it is said to be purely imaginary. The complex numbers p + iq and p iq are called Conjugate Numbers. The number p + iq is zero if and only if p = and q = 0. If p=p + iq } it is frequently found convenient to write B(p) for p and I(p) for q, where B()o) stands for the real part of p and I(/o) for the imaginary part of p. Complex Numbers are subject to the same algebraical laws of addition, subtraction, multiplication, and division, as real numbers. These operations, when applied to real and complex numbers, produce real and complex numbers only; and it will be shewn ( 6, 20) that this is also true of the remaining algebraical operation of root extraction. YA X' O Y' FIG. 1. 2. Geometrical Representation of Complex Numbers. The Complex Number s = x + iy can be represented geometrically by 2 FUNCTIONS OF A COMPLEX VARIABLE [CH. i means of a Rectangular Coordinate System X'OX, TOY (Fig. 1). The point P(x, y) corresponds uniquely to the number z, and is called the point z. In particular, points on the #-axis correspond to purely real numbers, and points on the ^/-axis to purely imaginary numbers. The figure is called the Argand Diagram, and the coordinate plane is spoken of as the 0-plane. Example. If z 1 and z 2 are conjugate numbers, shew that the straight line joining the points z 1 and 2 2 is bisected at right angles by the #-axis. 3. Modulus and Amplitude. In polar coordinates P is the point (r, 0), where r denotes the positive value of OP, and the angle XOP. The angle XOP is defined as the angle traced out by a radius-vector which revolves either positively or negatively from its initial position along OX till it coincides with OP. OP or r is called the Modulus of 0, and is written mod or | z \ ; is called the Amplitude * of 0, and is written amp 2. The amplitude can evidently have an infinite number of values differing from each other by multiples of 2-Tr: that value which satisfies the inequalities -7r<0^7T is called the Principal Value of amp z. The rectangular and polar coordinates are connected by the relations x = rcosO, y = r sin 0, r = Jx 2 + y\ tan = y/x. From these it follows that z x + iy = r (cos 4- i sin 0), an equation which expresses z in terms of its modulus and amplitude. Example 1 . Prove | cos + i sin 6 \ = 1 . ^ Example^. If z = x + iy, shew that |#|^|z| , \y |^| z\. Vectors. A line AB (Fig. 1), equal to, parallel to, and in the same direction as OP, may also be used to represent the number z ; mod (AB) and amp (AB) are then identical with z \ and amp z. AB is called a Vector. It follows that BA=-AB. * The word Argument is used by some writers in place of Amplitude. 3, 4] MODULUS AND AMPLITUDE 3 4. Geometrical Representation of Addition. Let P l and P 2 (Fig. 2) be the points z l = x l -{-iy l and z z = x z -\-iy z . Then YA R, FIG. 2. Through Pj draw P 1 P 3 equal to, parallel to, and in the same direction as OP 2 . P 3 has coordinates (fl? 1 + a? 2 , yi + y 2 ), and is therefore the point z^-{-z 2 . In vectorial notation OP = 3 = O?! + OP 2 = OP 2 Subtraction. Since X 2 = z l + ( 2 ), a subtraction can always be treated as an addition. Thus, if P 3 (Fig. 2) is the point % THEOREM I. The modulus of the sum of any number of complex quantities is less than or equal to the sum of their moduli : that is, if n is any positive integer, |^ + 2 +...+0 n ^l^l + l^l + .-. + l^n . This follows from the geometrical theorem that a side of a triangle is less than or equal to the sum of the other two sides : thus (Fig. 2) mod (OP 8 ) ^ mod (OPj) + mod (P^g). Therefore K + ^l^l^ | + K| . Hence z+2 + 2f = 2 + z + z and so on. THEOREM II. The modulus of the difference of two complex quantities is greater than or equal to the difference of their moduli. The verification of this theorem is left as an exercise to the reader. FUNCTIONS OF A COMPLEX VARIABLE [CH. i 5. Multiplication and Division. Let z 1 = r 1 (cos0 1 +isin0 1 ), z 2 = r 2 (cos 2 + i sin 2 ), . . . , z n = r n (cos 6 n + i sin Then, by De Moivre's theorem, , Hence, the modulus and amplitude of a product are equal respectively to the product of the moduli and the sum of Ihe amplitudes of the factors. In particular, if n is a positive integer, and if z = r (cos 6 + i sin 0), then z n = r w (cos nO + i sin 710). Example. If p + iq is a root of the equation a 2 w + a ] 2 M - 1 + ... + a n = 0, where the coefficients , a t , ... , a n are real, prove that p -iq is also a root. Again, ^ = ^ {cos (0 X - 2 ) + i sin (0 t - 2 )} ; %2 f% so that the modulus and amplitude of a quotient are respectively the quotient of the moduli and the difference of the amplitudes of the numerator and denominator. It follows that the equation z n = r n (cos nO + i sin n6) holds when n is a negative integer. In particular, mod (1/0) = I/ 10 1 and amp (1/0)= amp0. v Example 1. Give a geometrical construction for l/z. J Example 2. Shew that amp ( } = BAG. VAB/ Let OP and OQ be parallel to and in the same direction as AB and AC. Then amp (==} = amp AC - amp AB = ampOQ-ampOP = BAC. If the angle so obtained is a positive {Fig. 3 (a)} or a negative (Fig. 3(6)} reflex angle, the principal value of the amplitude of the quotient is obtained in the first case by subtracting and in the second case by adding 2?r ; the resulting amplitude is in the first case negative and in the second case positive. As a rule, when the amplitude is mentioned, it is to be understood that the principal value is referred to. 5, 6] MULTIPLICATION . / \ / \ */ Example 3. Shew that, if amp ( ^ ? ) = amp ( -? ) , the points 2 3 and 2 4 \z 1 z 3 / \Zi zJ are on the same side of the line joining z l and z 2 , and 1? 2 2 , ^ 3 , ^ 4 , are coney clic. Let Pj, P 2 , P 3 , and P 4 be the points z lt z 2 , z 3 , and 2 4 respectively. Then amp Therefore P 1 P 3 P 2 = P 1 P 4 P 2 . J^Ioreover, the points P 3 and P 4 must be on the same side of the line for if not, the angles P!?^ and P^^P^ would have opposite signs. Hence the points P 1} P 2 , P 3 , and P 4 are coney clic. Fio. 3. 6. Root Extraction. If n is a positive integer there are n distinct values of z n . For, since, if K is any integer, / O + ZKTT . . + 2/c7rY* . . 1 cos \-^ sin - ) = cos + ^ sin 6, \ n n / it follows that r n (cos + 2/C7T -fisin 2*-7T\ ) is I / an 7i th root of n n Now, if for AC the numbers 0, 1, 2, 3,..., i Ti1, are substituted in succession, n distinct values of z n are obtained. The substitution of other integers for K merely gives rise to repetitions of these values; and there can be no other values, since z n is a root of the equation x n = z, which has not more than n roots. Similarly, if p and q are integers, and q is positive, where K = 0, 1, 2, . . . , q 1. 6 FUNCTIONS OF A COMPLEX VARIABLE [OH. i / Example. Shew that the n th roots of any number can be represented by n equidistant points on a circle with centre at the origin. EXAMPLES I. 1. Shew that the straight line joining the points z l and z 2 is divided in the ratio m : n at the point (mz 2 + nz-^l(m + ri). 2. Prove that the centroid of the triangle whose vertices are z lt z 2 , and z 3 is(z 1 + z 2 +z 3 )/3. N/ 3. Prove that the modulus of the quotient of two conjugate numbers is unity. 4. Prove that amp z - amp ( - z) = TT according as amp z is positive or negative. v/5. If | z 1 1 = j z 2 1 , and amp2 1 + amp^ 2 = 0, shew that z l and z 2 are conjugate numbers. v * 6. If 2 cos = a + lfa, shew that 2 cos n6=a n + l/a n . ~~^~* /U 7. Prove algebraically that | z 1 +z 2 \ ^> \ z l \ + | z 2 \ . 8. Shew that, if \z 1 + z 2 + ...+z n \ = \z 1 \+ z 2 \ + ... + \z n \, the Z'B must all have the same amplitude. 9. Shew that, if amp | p ~ z v( z i ~ Z V j- = TT, then z 3 and z 4 are on opposite sides of the straight line joining % and z 2 , and z lt z 2 , z 3 , 2 4 , are coney clic. 10. Let A, B, C, and D be the points z lt z 2 , z 3 , and 4 . Shew that, if z^z 2 + 2 3 2 4 = and z 1 + z 2 = 0, then A, B, C, and D are coney clic and the triangles AOC and DOA are similar. 11. If AC : CB : : - AD : DB, and if A, B, C, D are the points z l5 z 2 , z 3 , z 4 , shew that A, B, C, an d D are coney clic, and prove (z l + z 2 ) (z 3 + z 4 ) = 2 (z^z 2 + z 3 2 4 ) : also prove triangles AOC and DOA similar, where O is the mid-point of AB. 12. Prove that the two triangles whose vertices are the points a l , a 2) a 3y and b lt 6 2 , 5 3 , respectively, are directly similar if and only if = 0. 13. Prove that the curves - constant and amp( -j = constant are orthogonal circles. ^/ 14. Prove that the imaginary n ih roots of a real quantity can be arranged in conjugate pairs. \/ 15. Picture on a diagram the roots of the equation s 5 + 1 = 0. 16. Shew that the equation 3 < 2,z = (z + l) 6 has four complex roots, two of which lie in the second quadrant and two in the third. Shew that all the roots lie on a circle. (See also Miscellaneous Examples, 1-9.) CH. ii, 7, 8] CHAPTEE II. FUNCTIONS OF A COMPLEX VAEIABLE. 7. Uniform Functions. When a variable complex quantity w is connected with another variable complex quantity z in such a way that to each value of z there corresponds one value of w, w is said to be a Uniform or Single-valued function of z. For example, a polynomial in 0, or the ratio of two polynomials, is a uniform function of z. The formal definition of a Holomorphic function of a complex variable will be given in Chapter III. The values of 0, for which w is a function of z, may be limited to some assigned region of the plane. Thus the equation where x is real, defines y as a function of x for those values of x and those alone which satisfy the inequality 1 < x < 1. Multiple-valued Functions. If several values of w correspond to each value of z, w is said to be a Multiple-valued or Multiform function of z. For example, >Jz is a two-valued, and V0 an ^-valued function of z. Path of Variation. In the theory of functions of a real variable, the independent variable x can only vary by values which correspond to points on the #-axis : in the theory of functions of a complex variable, on the other hand, the independent variable z can vary by values corresponding to the points of any path connecting the initial and final points. 8. Transformations. If w is a function f(z) of z, the relation between w and z may be interpreted geometrically, and the relation may then be called a transformation : the point z is said to be transformed into the corresponding point or points w by means of the transformation w=f(z). If iu = az + b, the trans- formation is called a linear transformation. If W<f>(z)/\fs(z), 8 FUNCTIONS OF A COMPLEX VARIABLE [CH. II where 0(0) and \fr(z) are polynomials, the transformation is said to be rational. Transformations of the type w = (az + b)/(cz + d) are known as bilinear transformations. We proceed to investigate the geometrical meaning of linear and bilinear transformations. I. w = z + b. Let P, Q, and B (Fig. 4) be the points z, w, and b. Then, since PQ = OB, it follows that the effect of the trans- formation is to impose on every point z a translation equivalent in magnitude and direction to OB. YJ 'P i FIG. 4. II. w = az. This transformation gives | w \ = \ a \ . \ z , and amp w = amp a -f- amp z. Consequently, if P and Q are the points z and w, the point Q can be derived from the point P by turning the radius-vector OP through an angle amp a and then multiplying it by | a \ . It follows that any figure in the plane is changed by the trans- formation into a similar figure. III. w = az-\-b. This, the general linear transformation, can be effected by applying transformations II. and I. in succession. X<ike transformation II. it transforms any figure in the plane into a similar figure. The ratio of the distances of corresponding points is given by the equation and the angle between corresponding lines by the equation amp(to 1 w 2 ) amp(0 1 z z ) = amp a. IV. w=l/z. Here |w]=l/|0f, and amp w = amp 0. Now 8] TRANSFORMATIONS let P (Fig. 5) be the point z and P' the inverse of P with regard to the circle [0| = 1. Then the modulus of P' is l/\z\ and its amplitude amp z. Again, let Q be the image of P' in the cc-axis ; then the modulus of Q is 1/|0|, and its amplitude is amp z. Hence Q is the point w. It follows that the transformation is equivalent to an inversion in the circle of unit radius with the origin as centre, followed by a reflection in the #-axis. Point at Infinity. As z tends to infinity, w approaches the origin. In the theory of the complex variable, infinity is regarded as a point ; namely, that point which is related to the origin by means of the transformation w=l/z. V. w = a/z. This can be regarded as a combination of trans- formations IV. and II. VI. The general bilinear transformation w = (az+b)/(cz+d), where a/b =f= c/d. (If a/b = c/d, then w is a constant.) This transformation can be written V (bc ad)/c 2 a It can therefore be effected by combining the three transformations z 1 = z + d/c, z 2 = k/z lt where Jc = (bc ad)/c z , and w = 2 4- a/c. It should be noted that z can also be derived from w by the bilinear transformation z = ( dw + b)/(cw a). Since the inverse of a circle is a circle or a straight line, O ' it follows that bilinear transformations transform circles into circles or straight lines. 10 FUNCTIONS OF A COMPLEX VARIABLE [CH. n V Example 1. Apply the transformation w = (22 + 3)/(2-4) to the circle Since w = 2 + ll/(z-4), the transformation can be effected by applying successively the transformations (i) ^ = 2-4, (ii) 22 = 1/2!, (iii) %=H%, and (iv) w= From transformation (i) we get #=#i + 4, y=^. Hence fo + 4) 2 + yi * - 4^ = 0. Transformation (ii) gives .Therefore 16(^ 2 2 +3/2 2 ) + 8^2 + 4y 2 + 1 =0. Again, from transformation (iii), so that 16 (> 3 2 + y 3 2 ) + 88^7 3 + 44y 3 +121=0. Finally, if w=u-}-iv, transformation (iv) gives #3 = ^-2, y 3 =-y. The given circle is therefore transformed into the circle I6u 2 + 16v 2 + 24w + 44v + 9 = 0. Example 2. Shew that the transformation of Example 1 changes the circle ^2+y 2 -437=0 into the line 4^ + 3 = 0, and explain why the curve obtained is not a circle. 9. Geometrical Representation of Functions. It is often convenient to represent the dependent variable w on a different plane from the independent variable z. This plane is called the u>-plane, and w = u+iv is represented on it by the point (u, v) referred to rectangular axes U'OU, V'OV. If w is a uniform function f(z) of z> and if z moves from a to b by different paths in the 0-plane, w will move from /(a) to /(&) by different paths in the w-plane. In the case of multiple-valued functions, how-- ever, it will be shewn that the final point attained in the tu-plane depends on which value of w is selected as initial value, and also on the path followed by z in the 0-plane. Example 1 . Let w = z\ so that u x 9 -- y 2 , v = 1xy . Then, if #=0, u= -y 2 and v = 0. Hence as z moves up the y-axis from - co to 0, u increases from oo to 0, and therefore w moves along the w-axis from - oo to 0. Again, as z moves up the y -axis from to +00, u decreases from to oo , and therefore ^ moves back along the -M-axis from to oo . Similarly, it can be shewn that as z moves along the #-axis from - oo to + oo , w passes along the w-axis from +00 to 0, and then back from to +00. Likewise, the positive and negative parts of the v-axis correspond respec- tively to the lines y x and y= x. 9] GEOMETRICAL REPRESENTATION 11 Again, if we put z=r(cos 6 + isin 6} and w = p(cos<f> + isin <), we have p = r 2 and < = 20. Hence, if z lies on the circle ABCD (Fig. 6) of radius a, w will lie on the circle PQRS of radius a 2 . Let 0=0, < = initially, so that A and P are the initial positions of z and w. Then as z passes round the quadrant AB in the anti-clockwise direction, Q and <f> increase to 7r/2 and TT respectively, so that plane A LU - plane FIG. 6. w passes round the semi-circle PQR. Similarly, it can be shewn that, as z passes round the quadrants BC, CD and DA, w passes round the semi-circles RSP, PQR and RSP respectively. Thus, when z describes the circle ABCD once, w describes PQRS twice. \/ Example 2. If w = z 2 , and if z describes the line # = c, shew that w describes the parabola u = c 2 -v*/4c 2 . Trace on a figure, for the particular case c = 1. the course of w as z moves up the line x = 1 from - oo . to +00. In applications it is often important to trace the change in the amplitude of w when z describes a closed curve. We shall consider some particular cases. Y M ( 1 ) w = z. Here amp w = amp z. Let z describe a closed curve LMN (Fig. 7) about the origin. Then, if z passes round LMN 12 FUNCTIONS OF A COMPLEX VARIABLE [CH. n once in the positive direction, amps, and consequently ampw, will increase by 27r. Similarly, if z passes round the curve once in the negative direction, amp z and amp w will each decrease by 2?r; while n successive revolutions in the positive or negative direction will alter the amplitudes by + Znir or 2mr. Again, if the origin is exterior to the closed curve APBQ (Fig. 8) described by 0, the amplitudes of z and w will increase Y from ^.XOA at A to ^XOB at B, and then decrease from 1.XOB to ^XOA; so that the total change is zero. (2) w = a(z z l ), where a and z l are constants. Here so that, since amp a is constant, the change in amp w is equal to the change in amp(0 zj. Hence, if z describes a closed curve surrounding z l in the positive or negative direction, amp w will alter by +27r or 2?r; while, if z 1 is exterior to the curve, amp w will return to its original value. In the first case w will describe a closed curve in the w-plane about the origin ; while in the second case it will describe a closed curve not enclosing the origin. (3) w = a(z %)( Zz)(z 3), where a, z v z 2 , and z 3 are -constants. Here amp w = amp a + amp (z X ) + amp (z z z ) + amp (z z 3 ). If z passes round the curve C (Fig. 9), which does not contain any of the points z lt z z , Z B , then ampw will return to its initial value ; so that w will describe a closed curve not enclosing the origin. If z passes round C^, C 2 , or C 3 , amp w will be altered by 27r, 47r, or 6?r, and w will pass round the origin once, twice, or thrice as the case may be. (4) w = a(z z^)(z z z )...(z z n ). If in this case z describes a closed curve within which none of the points 15 2 , ..., z n lies, 9] VARIATION OF THE AMPLITUDE 13 it follows, as in cases (2) and (3), that ampw will regain its initial value, and w will describe a closed curve which does not surround the origin ; while, if z describes a closed curve within which r of these points lie, amp w will be altered by 2r7r, and w will pass round the origin r times. (5) w = a(z-z 1 )/(z-z z ). Here amp w = amp a + amp (z z^) amp (z %). It follows that, if z describes the curve C x (Fig. 10) or C 2 in the positive direction, amp w is increased or decreased by 2?r ; while, O Fio. FIG. 10. if z describes either of the curves C or C 8 , amp0 regains its initial value. In all these cases it is obvious that the change in amp w due to the description of any closed curve is independent of the shape of the curve, so long as the same set of points z lt z 2 , z s , ... lies inside or outside it. It is often found convenient to take the curve in the form of a circle. (6) w = s/z. If z = r(cos + i sin 0), then w has two values, w^r^cos (0/2) + i sin (0/2)} and w 2 = r 1 / 2 {cos(0/2 + 7r) + 'i sin(0/2 + ?!)} = -w^ Each of these two quantities w l and w 2 varies with z, and is therefore a function of z : they are called the Branches of the two- valued function w. Let z start from the point P(r, a) (Fig. 11), and let the initial values of w 1 and w z be w = r 1 / 2 (cos (a/2) + i sin (a/2)} and w z = w r Then w^ and w 2 will be represented by the points P^r 1 / 2 , a/2) and P 2 (r 1 / 2 , a/2 -f TT) in the w-plane. Now, if z moves round the circle PQR of centre and radius r, 6 will increase by 2?r, and 14 FUNCTIONS OF A COMPLEX VARIABLE [OH. n amp w by TT. Consequently w l will move round the semi-circle PjQiRjP.2 and w z round the semi-circle P^R^ in the w-plane : the final values of w l and w 2 will be w z and w l . A revolution of 2 about the origin therefore interchanges the branches of w. Two such revolutions bring back w l and w z to their original values ; or, graphically expressed, if z moves round the circle PQR twice, w l and w z each move round the circle P 1 Q 1 P 2 Q 2 once. w- plane Q FIG. 11. If the circuit described by z does not enclose the origin, will regain its initial value a, and w l and w z their initial values w^ and w 2 . The point O is called a Branch Point of w, because a circuit about it interchanges the branches of the function. (7) w = -Ja(z z l ). Here amp w = \ amp a -f J amp (z X ). This is again a two-valued function. A single circuit about z 1 interchanges the branches, while a double circuit brings them back to their initial values. On the other hand, the description of a circuit which does not enclose z l effects no alteration in the branches. Hence z l is a Branch Point of w. . Here amp w = J amp a + \ amp(0 z ) + i amp (2 z 2 ). Hence the description of C x (Fig. 12) or C 2 interchanges the branches, while the description of C or C 3 leaves them unaltered. Thus 0J and 2 are Branch Points of w. (9) w y(z a). If z a = r (cos -f- i sin 0), w has n branches !/ 0+2S7T 9 + 2STT\ w,, w 9) .... w n , where w s r n (cos h^sm -I. A \ n n / positive circuit round the branch-point a increases by 2?r, and 9] BRANCH POINTS therefore changes u^ into w z , w 2 into w s , which do not enclose a leave the branches unaltered. 15 Circuits FIG. 12. Example 3. Let w = \/(l -z)(l +2 2 ), and let the value of w when z is at O be + 1. Then if z describes the curve OPA_(Fig. 13), where A is the point 2, shew that the value of w at A will be ~is/6. The three zeros of w are 1, i, and i. Let B, C, and D be the corre- sponding points, and through C and D draw CL and DM parallel to OX. Y C A X M FIG. 13. Let the moduli and amplitudes of BP, CP, and DP be r lt r 2 , r 3 , and < 1} </> 2 , < 3 , respectively, where z_XBP = ^> 1 , ^LCP = ^> 2 , and ^MDP = < 3 . Then / 1 \T "2^. ^,, It has still to be determined which of the two possible values +TT or -TT is to be assigned to amp ( 1). Now, when 2 is at O, <f> l = 7r, </> 2 = ?r/2, (/> 3 = 7r/2 ; so that < 1 + < 2 + < 3 = 7r. Hence, if amp(-l) = 7r, amp^=?r at O; while, if amp(-l)=-7r, ampw = at O; but 10= +1 when z is at O, so that the latter value must be chosen. Therefore Now, as 2 passes from O to A, ^ decreases from TT to 0, </> 2 increases from -7T/2 to -tan" 1 -!, and <j> 3 decreases from ?r/2 to tan" 1 .}. Therefore at A amp ?0= -7T/2 ; also ^ = 1, r. 2 =\/5, r 3 = \/5. Hence w = x/5 { cos ( - 7T/2) + 1 sin ( - ~/2) } - - i \/5. 16 FUNCTIONS OF A COMPLEX VARIABLE [OH. n 10. Roots of Equations. In works on the theory of equa- tions it is shewn how, by means of Sturm's Theorem, it is possible to find the number of real roots lying between any two real values of the variable. We shall now shew how to find the number of real or complex roots of an equation which are contained in various regions of the 2-plarie. Consider the equation We assume that every equation has a root : a proof of this important theorem will be given later (33). It follows that f(z) can be put in the form ^0 (Z-Zl)(z -%>... (2-Zn). If z be taken positively round a closed circuit in the 2-plane which encloses r of the points z lt z 2 , ... , z n , the amplitude of f(z) will be increased by 2r?r. Consequently the number of roots of f(z) = which lie inside a given circuit can be ascertained by det rmining the change in the amplitude of f(z) when z passes round the circuit. The following theorem will be found useful in locating the roots. THEOREM. If z be taken round any part of a large circle with the origin as centre and radius R, and if be the change in FIG. 14. amp z, the change in the amplitude of f(z) will differ from nO by a quantity which tends to zero as R tends to infinity. For f(z) = z n (a Q + ajz + a 2 /z* + . . . + a n /z n ). Hence amp/(0) = n amp z -\- amp (a + ajz + a 2 /z 2 + . . . + & n / 0n )- Now | ajz + a 2 /z* + . . . + a n /z 11 \ < p, where P = \ c^ j/R + | a 2 |/R 2 + ... +| On|/R n . Let R be chosen so large that /3<| I- Then the point ...+a n /z n must lie inside a circle of centre a or A ioi ROOTS OF EQUATIONS 17 (Fig. 14) and radius p. If OP be a tangent to this circle, amp(a +a 1 /2+...+a 7l /z n ) differs from ampa by an angle rj, which is not greater than ^AOP, and which can be made as small as we please by increasing R, and thus decreasing p. That is, amp/(0) = n amp z + amp a Y\. Hence Lim amp f(z) = n amp z -\- amp a . Therefore, when R tends to infinity, the change in amp/(z) tends to n times the change in amp z. Example. Investigate the positions of the roots of the equation Let w = 4 + 2 3 + l, and let z describe a contour consisting of the three portions : (1) the .t'-axis from to + QO ; (2) the first quadrant of a circle of centre O and radius infinity ; (3) the y-axis from +00 to 0. (1) At points on the .v-axis, w = u + iv=at + x? + l, so that u = and v = Q. Hence, as z passes along the .r-axis from to + QO , the w-axis from 1 to +00, and therefore amp w remains constant and equal to zero. (2) On the great circle ampz increases by ?r/2, and therefore, by the theorem above, amp w increases by 2?r. (3) At points on the ?/-axis, u=y* + l and v= -y 3 . Hence w lies on the infinite curve LMN (Fig. 15), given by these equations, and as y decreases VI FIG. 15. FIG. 16. from +00 to 0, w passes along this curve from infinity below the u-axis to the point M(w = l) in the direction indicated by the arrows. Hence the initial and final values of amp?0 are equal, both being zero. The total change in amp w as z passes round the complete circuit is there- fore 2?r, and it follows that one and only one root of the equation lies in the first quadrant. Similarly it can be shewn that only one root lies in each of the other quadrants. Again, let z describe the contour OABCO (Fig. 16), where A and C are the points 1 and 2', and ABC is a quadrant of the circle || = 1. 18 FUNCTIONS OF A COMPLEX VARIABLE [CH. II Then, firstly, the description of OA gives rise to no change in amp u\ Next, for points on a circle of centre O and radius R, z = E (cos $ + i sin 0) = R(l - P + 2&)/(l + * 2 ), where t = tan ((9/2), ' Accordingly, at points on ABC, z = (l + it)/(l it), so that Hence amp w = amp{(3- 12 2 + * 4 ) + z'(2 + 2 3 )}-amp(l -it)*. Now, as 6 varies from to ?r/2, t varies from to + 1 ; so that amp (1 - it) decreases by ir/4. Hence amp (1 it)* decreases by TT. Again, let = 3-12 2 + * 4 and ^ = 2^ + 2^. Then the curve given by these equations is of the form shewn in Fig. 17, FIG. 17. the arrows indicating the variation of the point (, rj) as t increases from oo to + oc . Now, when = 0, = 3 and 77 = 0, so that amp( + ir/) = ; also, when t = l y =8 and ?7 = 4, so that amp ( + 117) = 6, where is the angle in the second quadrant for which tan# = 1/2. Hence the change in arupw due to the description by z of the quadrant ABC is 7r+(9=2 7 r-tan-- 1 (l/2). Finally, at points on OC, u=y* + l and y= -y 3 , so that w lies on the curve LMN (Fig. 15). When y = l t wia at the point K(?0 = 2 i), and amp w tan -1 (l/2) ; while, when y = 0, w is at the point M(w = l) and arapw = 0: so that the change of amplitude due to path CO is tan -1 (l/2). Hence the total change of amplitude due to the circuit is 2?r, and there- fore the root which lies in the first quadrant lies within the unit circle. 10] ROOTS OF EQUATIONS 19 Similarly it can be shewn that the root in the fourth quadrant lies within the unit circle, while the other two roots lie outside it. Again, it can be shewn that all the four roots lie inside the circle |s| = 2. For, if z = 2(l+it)l(l-it), Now the curve 25 - 102 2 + 9 4 , 77 = 76* - 44Z 3 , is of the form shewn in Fig. 18, the arrows indicating the variation of the point (, 77) as t increases from oo to + 00. But as amp z varies from - TT FIG. IS. to +TT, t varies from - oo to +00 , and therefore amp(f + ITJ) increases by 4?r Also amp{l/(l -it)*} increases by 4;r. Hence ampw increases by STT, and therefore all the four roots lie inside the circle. EXAMPLES II. 1. If w and z are connected by the bilinear transformation and if the points u\ and ?/; 2 correspond respectively to the points z l and z 2 , shew that 2. If iv = (az + b)/(cz + d), and if the locus of z is an arc of a circle standing on the chord joining the points z l and z 2 , shew that the locus of w is an arc of a circle standing on the chord joining ^ and w 2 . 20 FUNCTIONS OF A COMPLEX VARIABLE [OH. 3. If w = (az + b)l(cz + d), and if the points w lt w 2 , w 3 , and u\ correspond respectively to z lt z 2 , z 3 , and 2 4 , shew that w 1 - ?/; 3 w 2 - w 4 z 1 4. Shew that the constants in the transformation w = (az + b)/(cz + d) can be so chosen that three arbitrary points w lt w^, and w 3 correspond respectively to three arbitrary points z lt z%, and z 3 . 5. Find the bilinear transformation which makes the points a, 6, and c in the z-plane correspond respectively to the points 0, 1, QO in the w-plane. z a bc A ns. w = -- T - . zc b-a 6. Find the bilinear transformation which makes the points 1, i, - 1 in the 2-plane correspond respectively to the points 0, 1, oo in the w-plane. Shew that the area of the circle z\=l is represented in the w-plane by the half- plane above the real axis. ATIS. w= i(z !)/(* + 1). 7. Prove that the relation w = (l+iz)/(i+z) transforms the part of the real axis between z=l and z= 1 into a semi-circle connecting w=l and w 1. Also find all the figures which, by successive applications of the relation, can be obtained from the originally selected part of the #-axis. 8. Let w=*J(2 2z + & 2 '), and let z describe a circle of centre z = l + i and radius \/2 in the positive direction. If z starts from O with the value \/2 of u\ what are the values of w (i) when z returns to O ; (ii) when z crosses the ^/-axis ? Ans. (i) - V2 ; (ii) V20 { cos (3:r/8 + A/2) + i sin (3;r/8 + A/2) } , where A is the angle in the second quadrant for which tan A= -3. 9. Let w = N /(5 - 2z + 2 2 ), and let z describe a circle of centre 3 = 1 + 2*' and radius 2 in the positive direction. If z starts from the point + 1 with the value +2 of w, find the values of w at the first and second crossings of the ^-axis. Ans. (i) \/2t/(20 + 8\/3){cos(7r/3 + a./2) + isin(Tr/3 + ./2)}, where a. is the angle in the second quadrant for which tan a.= 4 \/3 ; (ii) \/2^(20-8\/3){cos(27r/3 + ^/2) + isin (27r/3 + ^/2)}, where /3 is the angle in the second quadrant for which tan (3= -4 + >/3. 10. If vP=z+l, shew that, when the point z describes the circle |l=c, each of the points w describes the Cassinian ?v*. 2 = c, where r t and r 2 are the distances of w from the points +1 and - 1. 11. Shew that the equation 2^+2+ 1 = has one root in each quadrant, and that the root belonging to the first quadrant lies outside the circle ! z \ 1 and inside the circle | z \ = 2. 12. Shew that the root of z* + z + l =0 belonging to the first quadrant lies inside the square whose sides are ^ = 0, x\, y = 0, and y = l. 13. Shew that the equation 2 4 + 4(l + i>+l = has one root in each quadrant. n] EXAMPLES II 21 14. Shew that two of the roots of the equation z 5 -z + l6=0 have their real parts positive, and three their real parts negative. Also shew that all five roots lie outside the circle |^| = 1 and inside the circle |z| = 2. 15. Shew that the only root of z+ 1021=0 inside the circle z\ = l is real and positive. 16. Prove that z^ + lOz- 1 = has no root the modulus of which exceeds 2. 17. If w = { (2 + i)z + (3 + 4i) }/*, shew that : (i) as (x, y) describes the circle x 2 +y^ = l positively, the point (u, v) describes the circle (w-2) 2 + (v-l) 2 = 25 negatively ; (ii) as (#, y) describes the circle sfi+y 2 4. Qy 12=0 positively, the point (u, v) describes the circle (i*-l/2) 2 -f (v-13/12) 2 =(25/12) 2 negatively. 18. Apply the transformation w = l/z, (i) to the set of straight lines through the point (a, 0), and (ii) to the set of circles with this point as centre : and shew that the set (i) is transformed into a set of coaxal circles through the points w=0, w = l/, while the set (ii) is transformed into a set of coaxal circles, of which these two points are the limiting points. [CH. Ill CHAPTER III. HOLOMOKPHIC FUNCTIONS. 11. Limits. A single-valued function f(z) is said to tend to the limit L as z tends to the value z l if, corresponding to any assigned positive quantity e, however small, a positive quantity v\ can be found such that \f(z) L|<e for all values of z (except Zj) which satisfy the inequality | z z l | < 77. For brevity we write I L This condition can be represented geometrically as follows : if y (Fig. 19) be a circle in the w-plane of centre L and assigned radius e, a positive z - plane YI LU - plane X FIG. 19. quantity 77 can be found such that, so long as z remains inside the circle C in the 2-plane of centre z l and radius 77, the corresponding point f(z) in the w-plane will remain inside y. The limit L is clearly independent of the path by which z approaches z r The limit L has not necessarily the same value as /(^) : for, consistently with the definitions of 7, any arbitrary value can be assigned to the function at the point z r Limit at Infinity. If, corresponding to any positive quantity e, however small, a positive number N can be found such that 11, 12] LIMITS AND CONTINUITY 23 |/(0) L <e for |#|>N,/() is said to tend to the limit L as z tends to infinity : that is, Lim f(z) = L. Example. Lim 1 jz = 0. z >oo Infinite Limits. If, corresponding to any positive number N, however large, a positive number ^ cato be found such that \f(z) |> N for z l |< //, /(s) is said to/tend to the limit infinity as z tends to z r Example. Liml/0=co. The branches of multiple-valued functions generally tend to different limits as z tends to r If the limit L is a function L^) of z lt and if an rj can be found such that, for all points z^ in a given region, \f(z) L(0 1 )| <[ e provided | z z l \ < /, /(z) is said to tend uniformly to the limit L(0 1 ) in the region. 12. Continuity. The function f(z) is said to be continuous at z 1 if /(^) has a definite value, and if Lim f(z) = f(z^). If f(z^) is infinite, f(z) has not a definite value, and is therefore discontinuous at the point z : . The condition for continuity can be expressed as follows: if, corresponding to any e* an 77* can be found such that / -e for /(z) is continuous at z r A function is continuous in a region, if it is continuous at all points of the region. If f(z) has a definite limit at z l different from /(%), f(z) is said to have a Removable Discontinuity at z lt and the function can be made continuous by replacing the value at z l by the limit at that point. To investigate the continuity of a function at infinity, put = !/ and test for f=0. THEOREM 1. The sum of a finite number of continuous functions is a continuous function. * In this book e will usually be understood to represent an arbitrarily small positive quantity, and T? a positive quantity. 24 FUNCTIONS OF A COMPLEX VARIABLE [CH. m THEOREM 2. The product of a finite number of continuous functions is a continuous function. THEOREM 3. The ratio of two continuous functions is continuous except for values of z which make the divisor zero. The verification of these three theorems is left to the reader. The proofs are almost identical with those for functions of a real variable. THEOREM 4. If f(z) is continuous and has the value I at z lf and if <j>(z) is continuous at I, <f>{f(z)} is continuous at z r For, \f(z) l <, if \ z z i\<^*i', an d e can be chosen so that > if |f-J|O. Now let g=f(z); then provided | z z l |< >/. Hence [f(z)} is continuous at z = z v THEOREM 5. The real and imaginary parts of continuous functions are continuous functions. For, if w = u + iv is continuous at z = z lt and if its value at that point is w 1 = u l + iv l , an r\ can be found such that, for Thus J{(u>-u 1 y t +(v-v l )*\<e', so that | u u^ | < e, | v v l | < e. Hence u and v are continuous functions at z z v 13. Uniform Continuity. A function f(z) is said to be Uniformly Continuous in a given region, if, corresponding to any e, an 17 can be found such that, for every point z l in the region, \f(z) f(zj | < e, when \z %]<;; i.e. if /(0) tends uniformly to /(%) in the region. THEOREM. If f(z) is continuous in a given region, it is uni- formly continuous in that region. The proof of this theorem depends on the following Lemma : Lemma. If \f(z) -/K) |< e for \z-z l \<rj, then l/<X>-M)l<2e for |*-* 2 |<J* where 2 ^ 8 an y poi n t interior to the circle \z z l = | y. 13] UNIFORM CONTINUITY 25 Let G! (Fig. 20) and C 2 be the circles \z - z l = >/ and Then, if z and z 2 lie within C v !/(*) -M) I = !/(*) -M) +M) -/W I But if z 9 be restricted to lie within C 2 , every point z such that ^ J>; will lie within C r Hence the Lemma holds. Fir,. 20. Now suppose that f(z) is not uniformly continuous in a region within which it is continuous. Divide the region into smaller regions by means of sets of equidistant lines parallel to the two axes. In one, at least, of these smaller regions f(z) is not uni- formly continuous. Divide this smaller region into still smaller regions in the same way as before. In at least one of these regions f(z) is not uniformly continuous. By continuing this process a series of rectangular regions is obtained, each of which is contained in the preceding one, and is a region of non-uniform continuity. Now let z^ be a point interior to all the regions of this series; then, since f(z) is continuous at z l , an ^ can be found such that |/(0) /(%)(< e/2, provided \z z^\<r\. Hence, by the Lemma above, \f(z)f(z z ) <e for interior to the circle I z ], where z. 2 is \ ^rj\ so that this circle is a region of uniform continuity of the function. But if the subdivision of the given region be continued till a region of the series obtained above is reached which lies entirely within this circle, this rectangle is a region of non-uniform continuity for/(z). Also, since the rectangle lies within the circle, it is a region of uniform continuity for f(z). Thus two mutually contradictory results are obtained. Hence f(z) must be uniformly continuous in the given region. 26 FUNCTIONS OF A COMPLEX VARIABLE [OH. m Functions of Two Real Variables. A function u(x, y) of x and y is said to be uniformly continuous in a given region if, corresponding to any e, an tj can be found such that, for every point (x, y) in the region, \u(x + Ax, y + Ay)-u(x, y)\<e, provided Ax \ < 17, | Ay | < r\. A function u(x, y) which is con- tinuous in a given region is also uniformly continuous in the region. The proof of this theorem is left as an exercise, to the reader. Again, let the continuous function u(x, y) have continuous partial derivatives of the first order in the region. Then, if + Ay) u(x, y), u(x, y + Ay)} {u(x, y + Ay)-u(x, y)} ~\ r-\ u(x + O^x, y + Ay) + Ay O, y + 2 Ay), where 0<0 1 <1, 0<0 2 <1. Now and are con therefore, from the property of uniform convergence, Now and are continuous in the given region, and where a |< e, | /3 |< e, provided | Aa5 1< 17, | Ay |< jy. Hence Ait = ox where a and ft tend uniformly to zero with AOJ and Ay at all 'points in the given region. 14. Differentiation. The Derivative of any function f(z\ obtained by applying a finite number of the algebraical operations considered in 4, 5, and 6 to z in succession, is These limits are obtained by the same rules as when the independent variable is real. It is important to notice that the value of the derivative is independent of the amplitude of Az. 14, 15] DIFFERENTIATION 27 Example 1. Prove -^- =M2 li ~ 1 , (i) for a positive integer, and (ii) for n a negative integer. Example 2. Prove -j- = nz n ~ l for TI a positive fraction. Let n=p/q, where p and q are positive integers ; then, if z = r(cos + i sin 6), ~ / + 2rt?7r . . i/ -r ^-"- \ i 2=r(cos hesm 1, where & = 0, 1,2, ...,^-1. Now let i=z ll , where f represents the branch of z l/ * corresponding to one particular -value of k. Then, if the increment Az of z correspond to the increment Af of , Hence dz qf9-i q q where the same value of z l/<Jt is taken on both sides of the equation. Example 3. Prove -j- = nz n ~ l for n a negative fraction. 15. Holomorphic Functions. Any function of x and y can be regarded, according to the definition of 7, as a function of z : for if z be given, the corresponding values of x and y are known, and therefore the corresponding values of the function can be found. For example, one value of x iy or of x 2 y 2 corresponds to every value of z. But these functions cannot be expressed in terms of z, and it is much more satisfactory to regard them as functions of the two independent variables x .and y. Let w = u + iv, where u and v are real functions of x and y. Then, if z' = x iy, x = (z-\-z f )j% and y = (z z')/2i; so that u and v can be regarded as functions of the independent variables z and z'. Hence, if u and v are continuous f unctions % of x and y with continuous partial derivatives, the condition that w should be independent of z is 'du'dx , c. .'dv?>x .'dv 3 or _ _ -_ -cto J__ +1 %7 ~ and this is equivalent to the two equations 'du'dv ?>u cw 28 FUNCTIONS OF A COMPLEX VARIABLE [CH. nr Thus, if u and v are continuous, and possess continuous partial derivatives which satisfy equations (A), w is a function of x and y in which x and y occur only in the combination x-\-iy z\ it may therefore be expected that the function w, like the functions considered in 14, will have a derivative which does not depend at all on the way in which Ax and Ay tend to zero, i.e. which does not depend upon -f-. It will be shewn in the following doc theorem that this is the case. THEOREM. If iu = u + iv, where u and v are uniform con- tinuous functions which possess continuous partial derivatives, the necessary and sufficient condition that w should possess a definite continuous derivative is that these partial derivatives should satisfy equations (A). Let the increments Alt', Au, Av , and Az, of w, u, v, and z, correspond to the increments Ax and Ay of x and y. Then Aw _ Au + iAv Az ~ Ax -\-iAy Ax + i Ay where a, /3, a', /3', tend uniformly to zero with Ax and Ay. Thus div r . Aw - T -=Lim-: = , dz A2 _ >0 Az -, , .ay J- | t/ 7 eta; Hence the necessary and sufficient condition that -^-. should be 7 3 CtlJ independent of -r- is 'du . *dv _ 1 /du which is equivalent to equations (A). 15] HOLOMORPHIC FUNCTIONS 29 div 'dw 1 'dw COROLLARY 1. -?-==*-- dz ?)x i 3y COROLLARY 2. Since the partial derivatives of u and v are continuous, -j- is also continuous. az Definition. If a function is uniform and continuous, and possesses a definite continuous derivative at any point, it is said to be Holomorphic * at the point. A function is said to be Holomorphic in a given region, if it is holomorphic at all points of the region. Equations (A), expressed in terms of polar coordinates, become , ,. and the derivative is then obtained as follows : dw c)w 'dw "dr 'dw W sin = cos r +ir or \ or or./ r / A ' ^ W = (cos 6 i sm 6) ^ Oi Example 1. Shew that the function e x (cosy + i sin?/) is holomorphic, and find its derivative. Ans. e x (cos Example 2. Shew that logr + i'# is holomorphic unless r=0, and find its derivative. A ns. (cos 6 - i sin Q)/r. Note. From the definition of a derivative the rules for differentiating products and quotients follow as in the case of the real variable. THEOREM. If f(z) is holomorphic in a given region, then, for all points z l in the region, where A tends uniformly to zero as z tends to z r The proof of this theorem is left as an exercise to the reader. *The words Regular and Analytic are used by some writers instead of Holo- morphic. The sense in which we shall use the word Analytic will be explained in Chapter XII. 82. 30 FUNCTIONS OF A COMPLEX VARIABLE [CH. nr COROLLARY. If f(z) and <f>(z) are holomorphic, and if f(zj = 0, and 4>(z l ) = (\ while ^'( Function of a Function. If w=f(g) and =<f>(z) are holo- morphic functions of and z respectively, w is a holomorphic P (. dw dw d tunction of z\ tor ^ = - T? -f-. dz dg dz Simply-Connected Regions. If any two points in a region can be connected by a curve which lies entirely within the region, the region is said to be Connected. A connected region which is such that any closed curve lying entirely within it can be con- tracted to a point without passing out of the region is said to be Simply -Connected. Connected regions which are not simply- connected are said to be Multiply -Connected. The region enclosed by the curve C x (Fig. 21) is simply-connected, while the region between the curves C x and C 2 is multiply- connected. FIG. 21. The branches of multiple-valued functions can be treated as uniform functions in simply-connected regions which do not enclose any branch-points. No path in such a region can enclose a branch-point ; so that, after describing a closed path, the function regains its initial value. For example, -each branch of w = \/0 is holomorphic in the simply-connected region obtained by making the negative real axis a barrier which z cannot pass. Such a barrier is called a Cross-cut. The derivative l/(2/s), of course, takes the value corresponding to the value of *Jz under consideration. Inverse Functions. If w=f(z) is a holomorphic function such that w = w l corresponds to z = z l3 'z can be regarded as a function of w with z l corresponding to w l : if this function is uniform and continuous in a region of the w-plane which encloses w lt 15] INVERSE FUNCTIONS 31 then, since -y I -, -, it is a holomorphic function of w at all dw I dz , points of the region except those for which -y = 0. This function is called the Inverse Function of f(z). Example 1. If w = 2 , there corresponds to any value z l of z one value w 1 of w: conversely, one of the branches of z = *Jw gives the value z l of z corresponding to w = iv l . Now the only value of w for which -^-=0 is w = 0. Hence, if u\ + 0, w can be enclosed in a region in which the branch is holoraorphic, and therefore z = <Jw is the inverse function of w=z 2 . Example 2. For what values of z do the functions w defined by the following equations cease to be holomorphic ? (1) 2 = e"(cos v + i sin v) ; (2) z = log p + i(f>, where w = p(cos( + isin </>). Jws. (1) = ; (2) None. Laplace's Equation. It will be proved later (35) that if w = u-\- iv is a holomorphic function, w, u, and v have continuous derivatives of the second and higher orders. The reader can easily verify that u and v both satisfy Laplace's Equation The solutions of this equation are called Harmonic Functions, and are of great importance in Mathematical Physics. It follows that, if a function u or v is given, a corresponding holomorphic function w will not exist unless the given function is harmonic. If, however, this condition is fulfilled, the function w can be found by means of equations (A) : for example, if u is a uniform continuous function which satisfies Laplace's Equation, , 'dv .'dv -. is a complete differential, and v can therefore be found. Example. Shew that u=x z 3^/ 2 + 3.i i2 3?/ 2 +l is a harmonic function, and find the corresponding holomorphic function. A us. z* + 32 2 + 1 + ^C, where C is a real constant. Conjugate, Functions. If u + iv is holomorphic, u and v are called Conjugate Functions. These functions possess two im- portant properties : firstly, they satisfy Laplace's Equation ; and secondly, the curves u = c l , v = c 2 , where c and c., are arbitrary constants, intersect at right angles, since the product of their 32 FUNCTIONS OF A COMPLEX VARIABLE [OH. m ,. , . . mi. p gradients ( ~) and ( - / ) is 1. The systems of \dxj?iy/ \dxjcyy curves obtained by varying the constants c x and c 2 are called Orthogonal Systems. Example. Picture on a diagram the orthogonal systems given by 16. The Exponential Function. The function u+iv, where u+ iv = e x (cos y + i sin y) is holomorphic for all finite values of z, since u, v satisfy equations (A), 15. When y is zero, the function becomes the ordinary exponential function e x : it is therefore regarded as the extension of e x to the domain of the complex variable, and is denoted by fcxp (z). Obviously Again, since i sin y) x e x ' (cos y' + i sin y') exp (z) x exp (Y ) = exp (z + z). Hence exp (z) x exp ( z) = exp (0) = 1 ; so that exp ( z) = 1/exp (z). Thus, exp(z) satisfies the index laws. It is often found convenient to write e z for exp (3): in particular, e iy stands for cos y + i sin y. Derivative, -j- (expz) = {e x (cosy + i sin y)} = exp z, az ox *W dz Periodicity. Since cos y and sin y have the period 2?r, exp (z) has the period Ziir : i.e. e z+2kin = gz( cos 2/C7T + i sin 2/c?r) = e z , where k is any integer. Zeros and Infinities. Since e z e x , e z can only have zero and infinite values when e x is zero or infinite. But e x is only zero when x ~ oo , and only infinite when x = + oo . The Exponential Function is therefore finite and non-zero if x is finite. Example. Shew that every period of exp (z) must be an integral multiple of Ziir. io, 17] CIRCULAR FUNCTIONS 33 17. The Circular Functions. Since e ix = cosx + ismx and g - = cos x i sin x, e ixt e -ix e ix_ e -ix cos x ~ , sin 0;= ^- . 2 2t These functions can therefore be extended to the domain of the complex variable by means of the equations COS2 = 77 , Sin 5 = =-: , * 2i\> which define them as holomorphic functions. r riie following well-known formulae can be derived from these definitions : sm 2 2+cos 2 2 = l ; sin (2^ + 2 2 ) = sin z 1 cos 2 2 + cos z l sin 2 2 ; 008(0! + z 2 ) = cos z l cos z 2 sin z l sin z 2 ; d sin z d cos z .] = cos z 5 j = sin z ; sin( z}= sin z\ cos( z) = cosz. Note. If /( z) = f(z) for all values of z for which f(z) is defined, /() is said to be an odd function of z : if /( z) =f(z\ f(z) is an even function of z. Thus, sin z and cos z are odd and even functions respectively. Zeros. If sin 2 = 0, where k is any integer ; therefore is = iz + Skirl. Hence the values of z which make sin z zero are 0, TT, 2?r, 3-7r, ---- Similarly, since e- /z = e- l ' s +< 2 *+ 1 > ir *, the values of z which make cos z zero are given by z = (k + J)TT, where k = 0, 1, 2, ____ The other circular functions are defined by means of sin z and cos 2: e.g. tan = sin z/cos z. The inverse functions are written sin" 1 s, tan" 1 ^, etc. The Hyperbolic Functions. These functions are defined by the equations : e z + e~ z . . e z e~ z sinh z cosh z = 9 ; smh z ^ > tanh 2 = ^ ; etc. A . , , .. . , ., Example. Prove: -- -. = cosli2:; j =8inhz; cosh-i-sinli-r dz <(~- M.K. 34 FUNCTIONS OF A COMPLEX VARIABLE [CH. in 18. The Logarithmic Function. If y = e x , where x and y are real, the inverse function is x = log y : for complex values of the variables, the inverse of the Exponential Function is defined as follows : '^ j Let z = r*(co8 + i sin 9) = exp (w) = e"(cos v + i sin v}. Then e 1 * = r, so that i& = log r and v = + 2&7T, where & is any integer. Hence the inverse function is where may have an infinite number of values differing by multiples of 2-Tr. This function is denoted by Log z. If z passes round the origin once in the positive direction, increases by 2?r and Log z by 2i?r. The origin is therefore a branch-point of Log z. Each of the infinite number of branches of Log z is uniform and continuous in the simply-connected region formed by taking a cross-cut along the negative real axis; and therefore, since it satisfies equations (A') of 15, it is holo- morphic in that region. That branch for which TT < 5: + TT is denoted by log z ; for positive real values of z this branch is the ordinary Naperian logarithm. Zeros and Infinities. Since log r is infinite when r is zero or infinite, Log z has infinities at the origin and infinity. Log z is only zero when both log? 1 and are zero ; i.e. when 0=1. Derivative. ogg = e -** j- (logr + ifl) = J d * arV ?K ^.WV Example. Shew that Logos') = log z + log / + 2fe'. 1 g*> _ ~ ! '"' Function tan~ l z. If = tan w = - -.- -. , then ^ e lw +e~ M so that w = s~. Log f , where f = Now Log f is uniform if a 2i \%z cross-cut is taken in the f-plane* along the negative real axis. But the transformation =(l + iz)l(Iiz) is bilinear, so that one point in the -plane corresponds to each point in the s-plane, *The notation ^=^ + ir] is adopted ; , rj, f, then correspond to x, y, z. 18, 19] THE LOGARITHMIC FUNCTION 35 and conversely. Accordingly, if a cross-cut be taken in the 0-plane corresponding to the cross-cut in the -plane, the function Log f will be uniform in the s-plane. Now, since to the part of the -axis between and 1 corresponds the 2/-axis from i to +ix> , while to the ^-axis from 1 to oo corresponds the 2/-axis from ice to i. Hence, if a cross-cut is taken along these parts of the y-axis, the function tan~ 1 = ^ is uniform throughout the 3-plane. That branch which has the value zero when = is the Principal Value, and is equal to -r- -- -; its real part lies between Tr/2 and -rr/2, while - -- r- 1 ~~%'Z/ its imaginary part varies from oo to + x . For any other branch 1 . l where in is an integer. It follows that d 2, and that 1 19. The Transformation w = Log z. Since u = log r, to circles run^tt.int in the 0-plane correspond lines u = constant in the z - plane v> , , k, k. L, ^ U' v' VTT v=a Flo. 22. //-llane: the circles C , Cj, C. 2 , ... , C. lf C_ 2 , ... (Fig. 22) of radii 1 '. ' , ,..,6"^, e" a , ... correspond to the equi-distant lines L , Lj, L. 2 , ... , L_j, L_ 2 , ..., equations are ^6 = 0, 1, 2, ... , 1, 2, .... 36 FUNCTIONS OF A COMPLEX VARIABLE [OH. m To the origin and infinity in the z-plane correspond u oo and u = -h oo in the w-plane. Again, since v = 6, to the rays = constant in the s-plane correspond the lines v = in the w-plane ; so that, if a cross-cut be taken in the 0-plane along the negative #-axis, the entire s-plane is represented by that part of the w-plane which lies between the lines v = TT and v = + TT. If now the cross-cut be removed, and 6 increase from -w to STT, the entire 0-plane corre- sponds to the strip of the w-plane which lies between the lines v = TT and v = STT. Similarly the entire w-plane can be divided into strips of breadth STT, on each of which the entire z- plane is represented. Points in these strips which correspond to the same point in the -plane lie on the same parallel to the r-axis, at distances 2?r from each other. To each point in the i^-plane, however, corresponds only one point in the 2-plane, since exp (w) is a uniform function of w. Each strip of the w-plane represents one of the branches of w, the boundary in each case being assigned to the strip below it. Example. Shew that, for all values of m, - provided K( > 0. 20. The Generalised Power. Up to this point z n has only been defined for rational values of n ( 5, 6). We are now in a position to define it for all values of n, rational or irrational, real or complex. If w Log 0, then z = exp (w) ; hence z = exp (Log 0) = exp (log z + 2&7ri), where k is any integer. Accordingly, for all values of n, we ^define z n by means of the equation z n = exp (n log z + ZnltTri). COROLLARY 1. If n is an integer, z n lias only one value, exp (n log z), (cf . 5). COROLLARY 2. If n is a fraction pjq (q positive), z n lias q values given by exp ( log z] e**, where k = 0, 1, 2, . . . , q - 1 . ^\q The reader can easily verify that this agrees with the results of 6. 20, 21] CONFORMAL REPRESENTATION 37 COROLLARY 3. If n is irrational or imaginary, z n has an infinity of values. Example 1. Prove -i-=nz n - 1 for all values of n, where the same value of z n is taken on both sides of the equation. Example 2. Shew that, for all finite values of rx, ; > where s tends to infinity in any direction whatever. We have (15, Cor., p. 30) so that, if = l/, Lim 2 log(l + OL/S) = ou 2->OC Thus, since the exponential function is continuous ( 12, Th. 4), Lira ( 1 + - Y = Lim e* l <* < 1 + a ' 2 > = e. 2 _>oo \ Z J z _>. x See also Examples III. 13, 14, 15. 21. Conformal Representation. Let w be a holomorphic function of 0; then, if the points w t w lt W 2 (Fig. 23), in the w- plane correspond to the points z, z l} z 2 , in the z-plane, T . w,w dw r . Lim = ^- 2l ^ 2 z^-z dz T . w 2 w T . 0. 2 z or Lim = Lim . w l w z l z Hence, if the two triangles of vertices w, w lt w 2 , and z, z lt 0. 2 , w - plane z -plane FIG. '23. are infinitesimally small, they are directly similar. Also, since, to the first order of infinitesimals, x dw , (w l -w)= dz (z r -z\ the first triangle can be obtained from the second by turning it through an angle amp (dw/dz) and magnifying it in the ratio 38 FUNCTIONS OF A COMPLEX VARIABLE [CH. m \dw/dz\. It follows that two intersecting curves in the 0-plaiie are represented in the w-plane by curves which intersect at the same angle. Each plane is said to be represented Conformally on the other. Examples of Conformal Representation have been given in 9 and 19. The representation breaks down if -^- is either zero or infinite. Example. Deduce, from the principle of Conformal Representation, the theorem that the curves u = constant, v = constant, intersect at right angles, where u and v are Conjugate Functions. 22. Singular Points. A point at which a function ceases to be holomorphic is called a Singular or Critical Point, or a Singularity of the function. For example, 2 = is a singularity of 1/z. If a circle can be drawn with the singular point as centre, so as to enclose no other singularity of the function, the singularity is said to be Isolated. The function I/sin (1/z) has a non-isolated singularity at z = Q: for, since sin(l/0) is zero for z = l/(Jc7r), where k is any integer, it is impossible to surround the origin with a circle which does not contain an infinite number of these points. A point which can be made the centre of a circle enclosing no singularity is called an Ordinary Point. If the radius of the circle is equal to the distance of the point from the nearest singularity, the interior of the circle is called the Domain of the point. Poles. If Lim (z z^) n f(z) = C, where C is a non-zero constant n a positive integer, z 1 is said to be a Pole of f(z) of order n, and f(z) </>(z)/(z Zi) n , where <f>(z) is holomorphic at s r If n = l, z l is a Simple Pole of f(z). For example, l/z n has a pole of order n at z = 0. Example. The function I/sin (B ZI) has a simple pole at z l : for ( 15) Lim ! (z -,) 7 M tJ- s =H - 7 - it =1. ^V "sm (*-*,)/ \cos(* -*!)/, The function f(z) will have a singularity at infinity if =0 is a singularity of /(1/f). For example, az 2 +bz + c has a pole of 22] SINGULAR POINTS 39 the second order at infinity. If infinity is an isolated singularity of f(z), f=0 will be an isolated singularity of /(1/f), and a circle If | = e can be drawn to enclose no singularity of /(1/f) except f=0. Hence a circle \z =l/e can be drawn which will have within it every singularity of f(z) except infinity. Meromorphic Functions. A function which is holomorphic throughout a region except at isolated poles is said to be Mero- morphic in that region. Essential Singularities. If no value of n can be found such that Lim( z l ) n f(z) = C, then z l is said to be an Essential Singularity of f(z). Poles are Non- Essential Singularities. Example. The function e 1 -'-' has an essential singularity at 2 = 0. Branch-Points. The branch-points of multiple- valued func- tions are Singular Points : for example, z = Q is a singularity of Log z. Zeros. If f(z) = (z z l ) n (^(z\ where n is a positive integer and </>(z) is holomorphic and non-zero at z l} then z is said to be a Zero of f(z) of order n ; a zero of order 1 is also called a Simple Zero. If z l is a zero of f(z) of order n t it is a pole of l//(z) of order n. THEOREM 1. A pole is an isolated singularity. If z l is a pole of f(z) of order n, the function (z z l ) n f(z) is holomorphic at z l ; consequently, if C is its value at that point, an r\ can be found such that \(z z l ) n f(z) C\<^e ) provided | z z^ I < r\. Hence f(z) must be finite at all points except z l in the circle z z l \ = >/, so that the singularity is isolated. COROLLARY 1. The zeros of f(z) must also be isolated, or the function ~L/f(z) would have non-isolated poles. COROLLARY 2. If infinity is a pole of f(z), a circle can be drawn which encloses all the singularities of f(z) except infinity. THEOREM 2. No region can contain an infinite number of isolated singularities. Let a given region contain only isolated singularities, and let it be divided up as in 13. If there is an infinite number of singularities in the region, one at least of the divisions must contain an infinite number of singularities, and by continuing the process of subdivision a point can be found such that, in every 40 FUNCTIONS OF A COMPLEX VARIABLE [CH. m neighbourhood of it, there is an infinite number of singular points, i.e. it is a non-isolated singularity, which is contrary to hypothesis. COROLLARY. If a function is meromorphic throughout the plane, and has an ordinary point or a pole at infinity, it follows (Th. 1, Cor. 2) that it has only a finite number of singularities. EXAMPLES III. 1. Shew that l/{(z a)(z b)(z c)} is holomorphic except at , &, and c. 2. Shew that the following functions are holomorphic, and find their derivatives : (i) e~ y (cos#+isin#). Ans. ie~ y (cos x + i sin x). (ii) cosh # cosy + ^sinh#siny. Ans. sinh x cos y + i cosh x sin y. (iii) sin # cosh y + i' cos #sinhy. Ans. cos # cosh y z'sin^sinli?/. (iv) cos x cosh y i sin x sinh y. Ans. sin x cosh y i cos x sinh y. 3. If n is real, shew that r n (cosnd + isinn@) is holomorphic except possibly when r=0, and that its derivative is nr- 1 (cos n - 1 d + i sin n - 1 0). 4. For what values of z do the functions w defined by the following equations cease to be holomorphic 1 (i) z=e~ v (cosu + isinu). Ans. 2 = 0. (ii) z sinh u cos v + i cosh u sin r. Ans. z i. (iii) 2 = sin % cosh v + i cos w sinh v. Ans. z= 1. 5. If <f> and T/T are functions of x and y satisfying Laplace's Equation, shew that s+it is holomorphic, where * a ? B *5~ o^ an( i ^ ==: ^T^"^^' 6. Shew that w = e* (# cosy -y sin y) is a harmonic function, and find the corresponding holomorphic function. Ans. ze* + iC. 7. If w=z 2 , shew that the curves u = c lt v = c 2 , are rectangular hyper- bolas, and represent them on a diagram for different values of c and c 2 . 8. If z = sin u cosh v + i cos u sinh v, picture on a diagram the orthogonal systems uc^ v = c 2 . Shew that the first system consists of confocal hyperbolas, and the second of confocal ellipses. 9. Shew that, (i) sin iz = i sinh z, (ii) cos iz = cosh z. 10. Prove (i) sin (z 1 iz z ) = siuz l cosh z 2 i cos z^^ sinh z. 2 , (ii) cos (z l iz 2 ) = cos z\ cosh z. 2 T i sin 2 X sinh z. 2 . 11. Prove (i) | sin (x ty) | = *J (sin 2 # + sinh 2 y} = cosh y (ii) | cos (# + ^y) | = x /(cos 2 x -\- sinh 2 y} cosh y Sij sinhy |. 12. Prove Log ( - 1 ) = (2^- + 1 ) TT/, where -1 is any integer. 13. If w is real and z = re ie , shew that 2 n = r" e ni ^ +2{ ' n \ where r n is real and positive. 22] EXAMPLES III 41 14. Shew that z n z n = z" +n ' for all values of n and ri, where suitable branches of the functions are taken. 15. Shew that (z n ) n ' =z nn> for all values of n and ri, where suitable branches of the functions are taken. 16. If w={(z-c)/(z + c)} 2 , where c is real and positive, find the 'areas of the 2-plane of which the upper half of the w-plane is the conformal representation. Ans. (i) The lower half of the circle \z =c ; (ii) that part of the plane above the #-axis which is exterior to the circle | z \ = c. 17. If 10= -iccot(zf2\ shew that the infinite rectangle bounded by #=0, # = TT, #=0, # = 00, on the 2-plane is conformally represented on a quarter of the w-plane. 18. Shew that infinity is a simple zero of (az 2 + bz + c)/(lz 3 + mz 2 +nz+p). 19. Shew that the ratio of two polynomials is a meromorphic function. 20. Shew that sees, cosec?, taii, and cot?; are meromorphic in the finite part of the plane. 21. If w = sin -1 2, shew that w = /nr : Fi Log {1^ + ^(1 -z 2 )} according as the integer k is even or odd, a cross-cut being taken along the real axis from 1 to oo and from - x to -1 to ensure that Log{?^-f x /(l -z 2 )\ should be uniform. Deduce that 7 - sin" 1 z = --j- - ^ , dz V(l-s-) where the branch of -j- - is chosen which corresponds with the branch of sin- 1 .* under consideration. 22. Prove (i) Ljm-* i; (ii) Lta = - 2 ; z-n (-1)" , (in) Lim -- = , where n is an integer ; ' x T . tan A: . (iv) Lim -= A. c-0 - 23. Shew that all the values of i l are given by g-O^+iJ*, where / is any integer. 24. If w=\jz, shew that the curves it = c lt v = c t , are orthogonal circles which pass through the origin, and have their centres on the ?/-axis and ^r-axis respectively. [CH. IV CHAPTER IV. INTEGRATION. 23. Limit of a Sequence. Let z lf z 2 , z s , ... be an infinite sequence of real or complex numbers; the sequence is said to converge to a limit I if, corresponding to any assigned e, a number m can be found such that z n < e, when n ^ m. If z n x n + iy n a,udl = a + ib, then \x n a < e and y n b <e; hence it follows that the sequences x lt x 2 , ~x s , ... and y lt y%, y 3 , ... converge to the limits a and b. Conversely, if these two sequences tend to the limits a and 6, the ^-sequence tends to the limit a+ib. THEOREM. The necessary and sufficient condition that the sequence should have a limit is that, corresponding to any e, an n can be found such that z n+p z n \ < e, where p is any positive integer. This condition is necessary, for, if I be the limit, It is also sufficient, for it involves the conditions which determine the convergence of the x and y sequences. Uniform Convergence of a Sequence. It may be that all the z's are functions of a variable f : this is indicated by writing z n (g) for z n . Then if, at all points f in a given region, the sequence is convergent and has the limit (), and if an in can be found such that, for all points within the region, ^n(D"~^(DI < e when n^m, the sequence is said to converge uniformly within that region. 24. Curvilinear Integrals. Before defining definite inte^ grals of functions of a complex variable, we shall define curvi- linear integrals, and prove Green's Theorem. 23, 24] CURVILINEAR INTEGRALS 43 Consider a curve C (Fig. 24) joining two points A and B in the (x, y) plane. This curve can be divided into segments AL, LM, MN, . . . , such that, for each of these segments, only one value of y corresponds to each value of x ; and thus in each segment y is a uniform continuous function of x. Denote these functions by ^(x), <f> z (x), $%(%), Now let f(x, y) be a uniform continuous function of x and y in a region of the plane containing the path C. Then the FIG. 24. functions f{x, ^(x)}, f{x, <j> 2 (x)}, f{x, </> B (x)}, ..., are uniform continuous functions of x on the arcs AL, LM, MN, ... , respec- tively, and the integrals fl Cm Cn f{x, ^(x)} dx, f{x, </> 2 (x)} dx, f{x, </> B (x)} dx, . . . , a J I J m where a, I, in, ... t b, are the abscissae of A, L, M, . . . , B, are ordinary definite integrals. They are the Curvilinear Integrals J A M y) dx, J ^ f(x, y) dx, jjffa y)dx, ... , and their sum is the Curvilinear Integral \ f(x, y)dx. Jc Similarly, by dividing C into segments in each of which x is a uniform function of y, we can define the curvilinear integral I \lr(x, y) dy. By combining these a third type of curvilinear is obtained. 44 FUNCTIONS OF A COMPLEX VARIABLE [CH. iv COROLLARY 1. If x and y are uniform functions of a para- meter t, the integral becomes where t Q and t t correspond to the initial and final points of C. COROLLARY 2. If x and y are uniform functions of f and q, and if the curve T in the ( rf) plane corresponds to the curve C in the (x, y) plane, COROLLARY 3. If C be divided into n segments by points (x^ y^\ (x 2 , y 2 ), ... , (x n+1 ,y n+1 ), taken in order on the curve, where (a?!, T/J) and (aj n+1 , 2/ n +i) are the points A and B, and if (, i^), (2* %)' (f> tfnX are points taken at random on these seg- ments, the sum tends to the limit f {/(aj, y)dx+\js(x, y)dy} when the law of Jc division is made to vary in such a way that n tends to infinity and the greatest of the segments tends to zero. COROLLARY 4. f {f(v,y)dx + ^(x ) y)dy}=-[ {f(x, y)dx + ^(x, y)dy}. J BA J AB COROLLARY 5. If K is any point on C, f {f(x, y)dx+\f,(x, y)dy} = \ {f(x, y)da+^(x, y)dy} J AB J AK >, y)dx+\Is(x, y)dy}. COROLLARY 6. If C is a closed curve, the value of the integral is independent of the position of the initial point, but its sign depends on the direction in which the curve is described. Differentiation under the Integral Sign. If f(x, y, a) and - f(x, y, a) are continuous functions of x and y on the curve C, eta. 24, 25] GREEN'S THEOREM 45 and of the real parameter oc between assigned limits for oc, and if 1 J = 1 f(x, y, CL)dx, then <J>(OL) has a derivative given by c For = {/(# y> a+Ao.) f(x, y, a.)}dx. Jc Now, for points on C, f(x, y, oc) is a function of two variables x and oc ; hence ( 13) f(x, y, oc + Aoc)=/(a, y , oc) + |^/(^, 2/, aj where A tends uniformly to zero with Aoc. Therefore Aoc, and the latter expression tends to zero with Aoc. Thus 0(oc) has a derivative, given by 25. Green's Theorem. This theorem gives an important relation between a double integral and a curvilinear integral. Let the functions P(#, y) and Q(&, y) be uniform and con- tinuous, and possess continuous partial derivatives, in a simply- .V Fio. 25. X X connected region containing a closed curve C. Consider the taken over the simply-connected area enclosed by G. Assume in the first place that C (Fig. 25) is a curve such that 46 FUNCTIONS OF A COMPLEX VARIABLE [CH. IV no line parallel to either of the axes cuts it in more than two points. Let y l and y 2 (y% ^ y { ) be the values of y on C corresponding to any value of x, and let A and B be the points on C of minimum and maximum abscissae x and X . Then ~ If dxdy= ~ft 7(x ' &>- p <*' The latter expression is the sum of the two curvilinear integrals -f P(x,y)dx, \ P(x,y)dx; J AQB J APB and therefore, since I P(x, y) dx =\ P(x, y) dx, J AQB J BQA ~ J J ^ dX dy = J c P (X ' y) dX) the integral being taken round C in the positive direction. Similarly J J ^ dx dy = \ Q (x, y) dy. Hence Green's Theorem, holds for the region considered. Next, if C does not satisfy the condition that no line parallel to either of the axes cuts it in more than two points, the region can be divided into regions each of which possesses this property. For example, if in Fig. 26 the points A and B, at which the YJ o FIG. 26. tangents are parallel to the y-axis, are joined by a straight line, the two regions so obtained are of the type required. 25] GREEN'S THEOREM 47 Hence f((-^ + ! Q )<fo%=( (Pdx + Qdy)+\ (Pdx + Qdy) JJV ay ox/ JAQBA JABPA = { (Pdx + Qdy),. Jc since the sum of the integrals along AB and BA is zero. Thus the theorem can be shewn to hold for all simply-connected regions bounded by closed curves. COROLLARY. The area of the region enclosed by C is given by any of the three integrals Multiply -Connected Regions. Consider the region between the curves C and C' (Fig. 27). This region can be made simply- FIG. 27. connected by drawing a line LM from C to C'. Hence = \ J + Qdy), G J C' where the latter integral is taken positively round C'. Similarly, for the region between the curve C (Fig. 28) and the ?i curves c lt c 2 , c a , ..., c n , it can be shewn that 48 FUNCTIONS OF A COMPLEX VARIABLE [OH. Example. If Pd.v + Q,dy is a complete differential, she\y that D wlrere C is a closed curve. , I FIG. 28. 26. Definite Integrals. Let f(z) = u(x,y) + iv(x,y) be a uniform continuous function of z in a given region, and let ACB (Fig. 29) be a curve in this region connecting the points z and z. FIG. 29. Let z lt z z , ..., z n , be n points taken in order on this line, where z n is z, and let f x , f 2 , ..., f n be arbitrary points on the segments SUm 26] DEFINITE INTEGRALS 49 the real and imaginary parts be separated, we obtain where = Now if the law of division of the curve ACB varies so that n tends to infinity and the greatest of the segments tends to zero, this latter expression tends to the limit ( 24), I (u dx v dy) + i \ (vdx + udy), J ACB J ACB or I (u+iv)(dx + idy). JACB This limiting value of S n is called the integral of the function f(z) taken along the curve ACB, and is written JACB COROLLARY 1. From the theory of limits it follows that, corresponding to any e, an n can be found such that COROLLARY 2. f f(z)dz= - { f(z)dz. J BCA J ACB COROLLARY 3. [ f(z)dz~\- \ f(z)dz= [ f(z)dz. JAC JC'B JACB COROLLARY 4. {fi(z)+A( z ) + ~-+f n (z)}dz = f A(z)dz+ [ /()(&+...+[ f(z)dz. JACB JACB JACB COROLLARY 5. I kf(z)dz=1c\ f(z)dz, where A: is a constant. JACB JACB COROLLARY 6. I f(z)dz= \ /{0(f)} <}>'() d, where J ACB J ay ft s a holomorphic function of f, and the path ACB in the 2-plane corresponds to the path ocy/3 in the f-plane. M.F. D 50 FUNCTIONS *OF A COMPLEX VARIABLE [OH. iv For f(z)dz = I (u dx v dy) -f- i \(vdx + u dy) J ACB J J since z is a holomorphic function of COROLLARY 7. The modulus of the integral is finite. For, let M be the greatest value of \f(sm on ACB ; then, since where I is the length of ACB. COROLLARY 8. If F(z) is a holomorph c function whose derivative is f(z\ \f(z)dz = F(z)-F(z ). Jz For, let F(z) = U(x, y)+iV(x, y) ; then / Therefore I f(z)dz = \(u dx v dy) -f 'i \(v dx + udy) K3U Now the integrands are complete differentials; therefore the integrals are the limits of the sum of the increments of U(#, y) 26, 27] CAUCHY'S INTEGRAL THEOREM 51 and V(aj, y) obtained in going along ACB from (oj , y ) to (a;, y). Hence Since F(z) is single-valued, it follows that the value of the integral is independent of the path. Example. Shew that I z n dz=(z n+l -ZQ H+l )/(n + l) for all integral values J*9 of n except - 1. If n is negative, the path must not pass through the origin. See also Examples IV. 1-4. Consider now the integral I f(z)dz, where the path C goes to -/". Jc inimity. By means of the transformation z c = l/f, where c does not lie on C, C is transformed into a finite path C' with f=0 as final point, and the integral becomes In order that the integrand /(c + l/f)/f 2 should be continuous, Lim/(c-f l/f)/f 2 or ~Limz 2 f(z) must be finite. Hence the given f >0 z < integral has a definite value if Lim z 2 f(z) is finite. 2 >! ^Example. I -7= - J ^C = l> provided that the path in the z-plane does not go through the origin. 27. Cauchy's Integral Theorem. If a function f(z) is holo- morphic in a simply-connected region A, and if C is a closed contour lying entirely within A, I f(z)dz = 0. Jo 'Letf(z) = u + iv; then, by Green's Theorem, ff(z)dz=\ (udx vdy) + i\ (vdx + udy) c Jc Jc the double integrals being taken over the area enclosed by C. Hence (equations (A), 15) 52 FUNCTIONS OF A COMPLEX VARIABLE [OH. iv Example. From the integral / - , where C denotes the circle \z =1, deduce JcZ ' l+2cos0 T/1 The following important theorems are corollaries of Cauchy's Theorem : THEOREM 1. Let f(z) be holomorphic within a simply- connected region, and let the paths ABC (Fig. 30) and ADC 2-0 A FIG. 30. joining the points z and z lie entirely within the region. Then Jf(z) dz + I f(z) dz = 0\ ABC J CDA so that f(z) dz = \ f(z) dz. JABC JADC The integral is therefore independent of the path, so long as the path lies entirely within the region. THEOREM 2. Under the conditions of Theorem 1,F (z) = \ f(z)dz is a holomorphic function of z. Let the increment AF(0) of F(0) correspond to the increment A0 of z ; then fz+Az fz fz+Az AF() = /(*) dz - f(z) dz = /(f) df J ZG J ZQ ^ s |*z+Az fz+Az Iz+Az Now take A0 so small that l f()f(z) \ <e for all points f on the line joining z and z + Az ; then Therefore <. 27] CAUCHY'S INTEGRAL THEOREM 53 Hence F(z) has a definite derivative f(z) ; it is therefore holomorphic throughout the region. From this theorem the method of Partial Integration can be derived exactly as for the real variable. As in the theory of integrals of real functions, we say that a function F(z). which is such that ^-^ssf(z) t is an Indefinite Integral of f(z) ; and we write Example. Prove I log z dz = z log z z. THEOREM 3. Let/(z) be holomorphic in* the ring-space bounded by the curves C and C' (Fig. 27); then the integration in both cases being in the positive (or negative) direction. For, by Cauchy's Theorem, f f(z) dz + f f(z) dz - \ f(z) dz + [ f(z) dz = 0. Jc JLM Jc' JML But f /()<&= -f fa)dt. J ML J LM Therefore f f(z) dz = { f(z) dz. Jc Jc' Similarly, if f(z) is holomorphic in* the region between C (Fig. 28) and the n curves c lt c 2 , ..., c n , it can be shewn that THEOREM 4. If a is a point enclosed by a curve C, tdz = 27Tl. c z a Round a describe a small circle c of radius r ; then (Theorem 3) f dz _f dz Jc^'Jc^V * Here, as in Cauchy's Theorem, it is to be understood that the boundaries lie inside a region in which /(z) is holomorphic. 54 FUNCTIONS OF A COMPLEX VARIABLE On c let z a = re ie ; then dz = re ie i dO. Therefore [CH. IV G z-a 28. Cauchy's Theorem : Alternative Proof .* The following proof of Cauchy's Theorem does not depend on Green's Theorem. The proof will be taken in three parts : firstly, for C a triangle ; secondly, for C an arbitrary polygon ; and, lastly, for C any closed curve. (1) Let C be a triangle A (ABC in Fig. 31), and let the mid- points D, E, and F of the sides be joined, so that the triangle is divided into four congruent triangles A', A", A 7 ", A iv . D FIG. 81. Now integrate round these four triangles in succession in the same (positive) direction, as indicated by the arrows. The two integrals along each of the lines DE, EF, and FD cancel each other, so that the net result is the integral round A in the positive direction. Hence - [ /(*)<fo=f /(*)<**+[ f(z)dz+\ f(z)dz+\ f(z)dz; JA JA' JA" JA'" J A lv so that f(z)dz f J z)az There must therefore be at least one of these smaller triangles we denote it by A x such that f(z)dz Cf. Knopp, Funktionentheorie, Vol. I. 27, 28] CAUCHY'S INTEGRAL THEOREM 55 Dealing with A a in the same way, we obtain a triangle A 2 such that [f(z)dz ^4 \f(z)d J A! J A 2 \f(z)dz J A and therefore Proceeding thus, we obtain a sequence of similar triangles A, A!, A 2 , ..., each contained by the preceding one, and such that i-A n , and \f(z)dz ^4 J As n tends to infinity, the triangle A n shrinks to a point, f say, which lies within every one of the triangles A, A x , A 2 , Now, corresponding to any e, an y can be found such that | X |< c if z f | < >7, where Let ti be chosen so great that A n lies entirely within the circle | z f | = v\ ; then f f(z)dz= \ f(Qdz+ f f JA JA JA The first two of thes integrals vanish ( 26, Corollary 8), so that ., ., f(z)dz=\ \(z-)dz. J An J A,, Therefore If f(z)dz ^f \\\\z-\\dz\ I J A u J A,, O = -jc8 n , where s n is the perimeter of A n , Hence Therefore e / -s 1 \' 2 . } ( ^j , where s is the perimeter of A. f f(z)dz = 0. JA 56 FUNCTIONS OF A COMPLEX VARIABLE [CH. iv (2) If C (Fig. 32) is a quadrilateral, it can be divided into two triangles A and A' by a diagonal which lies within it ; then f f(z)dz=\ f(z)dz+\ f(z)dz = 0. Jc JA JA' Similarly, if C is any polygon, it can be divided into triangles FIG. 32. by diagonals lying within it, and, since the integrals along these diagonals cancel each other, I f(z) dz = 0. Jc (3) Let C be any closed curve ; then, as in 26, < n f(z)dz = LimS n> where S n = ^f(z r )(z r -z r . l ). JG r=i Now let f(z) =f(z r ) 4- r] r f or points z on the straight line joining z r -i and z r , and let the law of division of C vary so that, for r=l, 2, 3, ..., n, |^ r |<e/(2L), where L is the length of C, and also Then, if P is the polygon of vertices , z lt z 2 , ... , z n , \ /(*> <fo = s r </w + *> dz = s + s r *** JP r = \Jz r -i r = lJ2 r _! But f(z)dz = Q; therefore Jp 11 Therefore Jo" Hence 28, 29] C^JCHY'S RESIDUE THEOREM 57 29. Cauchy's Residue Theorem. If the point a is the only singularity of f(z) contained in a closed contour C, and if V has a value, that value is called the Residue of f(z) at a. From Theorem 3, 27, it follows that, if encloses several singularities, the sum of the residues at these points is The following cases are important : CASE 1. If n is any integer except 1, the residue of (0 a)~ n at a is zero. CASE 2. The residue of (z a)~ l at a is unity. .' CASE 3. If f(z) = &J(z - a) + A 2 /(z - a) 2 + . . . + A n /(z - a) + 0(), where 0(2) is holomorphic at a, the residue of f(z) at a is A r Example. Shew that the residue of (2s + 3)/(s I) 2 at 1 is 2. CASE 4. If f(z) is holomorphic at a, the residue of f(z)/(z a) at a is /(a) : for i ( f( Z ) _ i r /(a) i r X c Now take C so small that for all points on it X <e; then where i is the length of C. Hence -Uf -dz=f(a). ZtnJQZa This is equivalent to the following theorem : THEOREM 1. If Lim {(z-a)f(z)} is a definite number A, the z >a residue of f(z) at a is A. 58 FUNCTIONS OF A COMPLEX VARIABLE [OH. iv It follows that, if <f>(z) and \fs(z) are polynomials in z, and if z a is a non-repeated factor of \Js(z\ the residue of <f>(z)/\ls(z) at a is <f>(a)l\l/(a). Example. Shew that the residues of (pz 2 + qz + ?)/{ (z - a) (z - b) } at a and b are ( pa 2 + qa + r)/(a - b) and (pb 2 + qb + r)/(b - a). See also Examples IV. 5-7. Multiply-Connected Regions. If f(z) satisfies the conditions of Theorem 3, 27, except for isolated singularities at points in the space between C and the curves c l} c z , ..., c n , the sum of the residues at these points is Residue at Infinity. If f(z) has an isolated singularity at infinity, and if C is a large circle which encloses all the singu- larities of f(z) except infinity, the residue of f(z) at infinity is defined to be taken round C in the negative direction (negative with respect to the origin), provided that this integral has a definite value. If the transformation 2=l/f be applied to the integral, it becomes taken positively round a small circle about the origin. Hence it follows that, if Lim { /(!/)/} or ^^ m { ~ z f( z )} nas a definite value, that value is the residue of f(z) at infinity. Example. Shew that the residues of z/{(z-a)(z-b)} and (z 3 - at infinity are - 1 and 1. Xote. Both of these examples shew that, if a function is holomorphic at infinity, it does not necessarily follow that its residue there is zero. THEOREM 2. If a uniform function has only a finite number of singularities, the sum of the residues at these singularities, that at infinity being included, is zero. Let C be a closed contour enclosing all the singularities of f(z) except infinity: then the sum of the residues at these singularities is I 29, 30] CONTOUR INTEGRALS But the residue at infinity is 59 Hence the sum of the residues is zero. Example. Evaluate the residues of z*/{(z- l)(z- 2)(2-3)} at 1, 2, 3, and infinity, and shew that their sum is zero. Ans. 1/2, -8, 27/2, -6. See also Examples IV. 8-10. 30. Evaluation of Definite Integrals. Many definite in- tegrals can be evaluated by means of integrals round closed contours. Example 1. Prove T cos x d * = , where >0. Jo x't + a' 1 2a Integrate f(z) = e* z /(z* + a 2 ) round the contour (Fig. 33) consisting of : (1) the A'-axis from R to R, where R is large ; (2) that half of the circle ] z \ = R which lies above the #-axis. O R X Fio. 33. The only pole of f(z) within the contour is ia, at which the residue is Li m \(z-ia z-^ia \ e ~ a 55" Hence / /(.,-) But ^c 1 P TrR Hence the integral along the semi-circle tends to zero aa R tends to infinity. Therefore so that f" - S>Y dx- v *~* Jo x* + a* 2a ' 60 FUNCTIONS OF A COMPLEX VARIABLE [CH. iv THEOREM 1. Let AB (Fig. 34) be that arc of the circle z\ = E for which 0^0^ 2 , where = amp3; and let zf(z), as R tends FIG. 34. to infinity, tend uniformly to the limit K, where K is a constant ; then f Lim JAB For, let zf(z) = K + X, and choose R so great that X | < e ; then f K + X . IP 2 = \ 9 \dO <e(0 2 -0 1 ). Hence Lim f(z) dz = i(0 0,)K R->< J^ AB For example, in Example 1, | zf(z) \ ^ R/(R 2 - a 2 ), so that K = 0. Example 2. If m > 0, prove T Jo cos mx dx TT - V m ; [Integrate e imz /(l+z 2 + z*) round the contour of Fig. 33.] From Theorem 1 it follows that, if f(z) = (j>(z)/ifs(z), where \fs(z) is a polynomial of degree n, and 0(0) is a polynomial of degree less than n, Limf f(z)dz = i(0^0 l )^, R^^JAB o where a and b are the coefficients of z 11 ' 1 and 2 n in 0(2) and \fs(z) respectively. In particular, if the degree of </>(z) is ^n 2, a is zero, and therefore the integral of f(z) round the contour of Fig. 33 gives r: 30] WORKED EXAMPLES 61 where 2 denotes the sum of the residues of f(z) at points above the #-axis. Example 3. Prove / / x = v ' , where a > 0. Jo XT + a* 4er The residue of l/(2 4 + 4 ) at a pole a. is But the poles above the #-axis are ae ln /* and ae l37r / 4 . Therefore |_ _^_ i = 2 M A e-' : Hence / -^-.-^. The inequality sin^^20/7r, where 0^0^7r/2, is frequently found useful. Example 4. Prove J - 2 ' ^ dr = ^ e~ a , where a > 0. Integrate /(2)=ze <z /(,2 2 + a 2 ) round the contour of Fig. 33. The only pole within the contour is m, the residue at which is e~ a /2. Hence f m J-K /W But r f(Re)Reide ^ I *T^"'7 dO 2R 2 2 2 - 2 a ./ o JR 2 ri ^R 2 2R 2 f 2 -a 2 Jo TrR Hence Lim [V(Re ie )R^'^<9=0. Thei'efore / y - so that r^ sil >'o ^ 2 4 >. Integrate e'* over the following contour (Fig. 35) : O R X Fin. ;;;.. (1) the .f-axis from <) to I! ; 62 FUNCTIONS OF A COMPLEX VARIABLE [CH. iv (2) the circle |2| = R, from 0=0 to #=<x, where . j ; (3) the line #=., from |s| = K to O. When R tends to infinity (1) gives I e~ x<i dx or . / ^ On (2)e-z 2 =e- R2 cos20 e -iR2sin20 ; so that, if 20=7r/2-<, *dz = e ~ & si d<, where /? = Tr/2 - 2o. Accordingly, when R tends to infinity, this integral tends to zero. Again, on (3)z=re la , and therefore, when R tends to infinity, thejjntegral becomes r 00 - e ~ ' C08 2a { cos(r 2 sin 2cx) - i sin (r 2 sin 2a.) } e la -dr. Jo But the integral is holomorphic within the contour ; hence /" e- J(\ E. (cos rx - 1 sin (/.). Therefore, if the real and imaginary parts are equated, I e~ x " cos 2<x cos(# 2 sin 2o.) dx ^- cos a., Jo 2 and / ~ x2cos2a sin(.r 2 sin2a.)o?^=-^sina.. Jo 2i If a.=7T/4, these integrals become the Fresnel Integrals r 2 r ^^ I cos SV* d\v ~^ \ sin '?'*"' ct-& == ~ A Jo Jo 4 Example 6. Prove / --J = -r-^ , where < a < 1. Integrate f(z) = e az /(I+e z ) round the contour (Fig. 36) consisting of the Y, 27TZ 1 _ "It -R O Fie ~R . 36. .r-axis and the lines x R, y = 27r. The only pole within the contour is TTZ, at which the residue is - e a7ri . 30] CONTOUR INTEGRALS 63 If *=R + z)/, then \f(z) \ ^ e nR /(e* - 1 ), so that Lim/(*)=0; hence the R >oo integral along #=R vanishes when R tends to infinity. If z=-R + iy, then | fts) ^ e -" R /(l -e~ R ), so tha,t Lim /"()= 0; thus the R->oo integral along x= R vanishes when R tends to infinity. 1 f Therefore ^j Hence Therefore The transformation 6^=^ changes this integral into l+y sm TT Two methods can be employed to evaluate integrals of the type I /(cos 0, sin 0) clO, when f(x, y) is rational in x and y. J -7T The first is to use the transformation # = tan ^0. The integral {+00 R,(x)dx, where R(a?) is rational in x. The alternative method is to apply the transformation z = e ie , and integrate round the circle \z = 1 . Example 7. Prove / - ^ ; = -^"~ , , where the sign of *J( 2 -b z ) J a + bcosti ^/(*-6*) is chosen to satisfy the inequality | a - *J(a' 2 - b 2 ) \ < \ b \ ; it is assumed that a/6 is not a real number such that - 1 =a/b^. 1. !*= ' d dz dz - ib J c (z - - Jo + 6cos(9 i c bz* + 2az + b ib c (z - OL)(Z - / where C is the circle \z\ =1, and a. and /? are the roots of Since ./? = !, it follows that either |o. or \/3\ is less than 1, or that 1 a. | == | ft I = 1. The latter alternative is excluded, however, since in that case a/6 would be real and such that -l^a/6^1. Let a.=(-a + Ja*-b'*) /6, where the sign selected for \/ 2 - b' 2 is that which makes | a. | < 1. Then THEOREM II. If Lim(z a)f(z) = K, where A: is a constant, fz->a f(z)dz, the integral being taken round the arc ^O., of the circle z a\ r, is i^ O^K. 64 FUNCTIONS OF A COMPLEX VARIABLE [OH. iv For, corresponding to any e, an r\ can be found such that if | z a < rj, A | < e, where (z a)f(z) = K + A. Hence, since Therefore Prove Lim (f(z) dz = i(0 2 - /- >o J f'iiEf^^f. Jo # 2 Integrate f(z) = e iz /z round the contour (Fig. 37) consisting of (1) the ,^-axis from r to E, where r is small and R large (2) the upper half of the circle | z \ = R ; (3) the .r-axis from R to r ; (4) the upper half of the circle \z\ = r. -R R X FIG. 37 Let I be the integral duetto (2) ; then =f< iRcosfl E sinfl n i Hence e de Therefore Lim I = Q. R-o Again, Lim2/(2)=l, so that (Theorem II.) the integral along (4) tends to ITT as r tends to zero. Hence Lim | J e -dx+ ^ ^-ctoj=7n. Therefore \\ T hen, in the description of a contour, part of a small circle is described to avoid a singularity of the integrand, the contour 30] PRINCIPAL VALUE 65 is said to be 'indented' at the singularity: for example, the contour of Fig. 37 is the contour of Fig. 33 indented at O. Example^. If 0<v< 1, prove / - dx=- o = 2 p ~" 1 /(l +z) round the contour (Fig. 38) consisting of : (1) the ^-axis from r to R ; (2) the large circle | z | = R ; (3) the .?;-axifi from R to r ; (4) the small circle | z \ T. FIG. 38. Within this contour f(z) is uniform. Consider that branch for which amp 2=0 on (1). Since p >0, Lini2/(2) = : hence the value of the integral along (4) tends *-*o to zero as r tends to zero. Again, when \z = R, 2/(2)|^R p /(R 1) : therefore, since jo<l, the integral along (2) tends to zero as R tends to infinity. At the point 1 amp 2 = 77: hence the residue at this point is eCf--*)**. Also on (3) amp ,s = 27r. Therefore r^B-l cfe*ftrttfP- 1 K 1 -\-x Hence f^^^' . Jo \+x sinpTr The substitution x = e y transforms this integral into Principal Value of an Integral. If /(z) is holomorphic in a region containing that part of the ic-axis for which a = x = b, except for a simple pole at a point c on the #-axis, where a < c < 6, then " tl-nds to a definite limit as e tends to zero. 66 FUNCTIONS OF A COMPLEX VARIABLE [CH. iv r& ^7/v, = log (b c) log (c a). cfce T . Hence Lim Now, let/(s) = 0(2)/(2-c); then ( 15, Theorem, p. 29) where X is continuous in the region. Therefore Lim { f f(x)dx + i f(x)dx\ e->0 Ua Jc+e J This limit is called the Principal Value of I f(x)dx, and is ritten - 6 P /(aj)do;. Ja Example 10. If < a < 1, prove rx a ~ i - dX = 7T COt 7T. 1-.^ Integrate z?~ l /(z-l) round the contour of Fig. 38 indented at 1 (Fig. 39). Example 11. If TT < a < TT, prove rsinh ax 7 1 a -^ dx -^ tan - smh Tr.r 2 2 Integrate e a2 /sinh (TTS) round the rectangle (Fig. 40) of sides ?/ = 0, y = l t x= R, indented at O and' 2. 30,31] THEOREM ON RESIDUES 67 Example 12. Integrate e ibz l(r-\-iz) a , where 0< < 1, r >0, b >0, round O FIG. 40. R X the contour of Fig. 41, where it is assumed that amp (r+ iz) is zero at points on the i/-axis between and ir ; and thus prove - 4* 27T ,_,_, r r; Prove also f + * *'** =0 J and shew that o PIG 41. If r= 1, .r = tan ^, deduce F (cos 6) a -' 2 cos ad cos (6 tan 6}dd '*T(a) ( 31. Theorem. Let C be a closed curve such that f(z) is holo- inorphic within and on C and 0(0) is meromorphic within and has no singularities or zeros qn C ; then where n 1 , a.,, a :i , ... are the zeros of 0(0) w r ithin C of orders r i r 2' r s-" respectively, and 6 1? 6 2 , 6 3 ,... are the poles of <j>(z) within C of orders s p *.,, ,s.,, ... respectively. 68 FUNCTIONS OF A COMPLEX VARIABLE [CH. rv For (p(z) = (z a l ) ri \Is(z), where \fs(z) is holomorphic at a x ; hence so that <p(z) z-a, The residue of the integrand at a x is therefore Similarly, since (2? 6 1 )* 1 ^() = x( X where x(^) i- s holomorphic at b lt the residue at 6 X is Hence OOBOLLA.T1. OOBOLLABY 2. Example 1. If ^>(^) is a polynomial of degree , shew that 2^ = ?^. Example 2. If <^(*) is a polynomial with factors 0., z /3, ..., shew that oc~* : +)8~ A: +...= -B, where /: is any positive non-zero integer, and R is the residue of -i$fe) a ts = 0. *^() 32. Liouville's Theorem. A function which is holomorphic at all points of the plane, including infinity, must be a constant. Let f(z) be such a function; then, if a and b are any two distinct points, the only singularities of the function are a and &, and possibly infinity. But since LimzF(2) = 0, the .-> ce residue of F(z) at infinity is zero ( 29, p. 58). Now the sum of all the residues is zero ( 29, Theorem 2) : hence so that /(a) =/(&) ; and therefore, since a and 6 are arbitrary points, f(z) is a constant. COROLLARY. Every function which is not a constant must have at least one singularity. 31-34] DIFFERENTIATION UNDER INTEGRAL SIGN 69 33. The Fundamental Theorem of Algebra. If f(z) is a polynomial in z, the equation /(z) = has a root. For, if not, the function I/f(z) would be finite and holomorphic for all values of z, and would therefore be a constant (Liouville's Theorem). Hence f(z) would be a constant, which contradicts our hypothesis. 34. Differentiation under the Integral Sign. Let the func- tion f(z, f) of the two independent complex variables z and f be holomorphic with regard to both z and f so long as z lies in a region A of the z-plane and in a region A' of the f-plane. Then the function #().= I f(z, )dz, where C lies entirely in A, is holo- f 7^ morphic at all points of A', and 0'(f ) = I ^/(z, f ) dz. Let f(z, ) = u-\-iv and 0() = P-f- / iQ, so that P = I (udx v dy), Q = I (vdx + u dy) ; then ( 24), 3P_f Cdu, 'dv_, \ 9P_f fdu 3?~JcW 3? y )' ^~Jr 3Q f /9v 7 , 9^ 7 \ 3Q f ^l = V^^ + ^^2/J' ^ = Sf Jc^f 3 y ^ J Hence (equations (A), 15), Thus 0(f) is a holomorphic function of f : its derivative is given by JExample. Integration under the Integral Sign. Shew that, if C' and C' lie in A and A' respectively, ( ff(z,C)dzd{=[ i f(z,t)d{dz. Jc f Jc Jc Jc f Let {" and f be the lower and upper extremities of C'; then/ / f(z,t)ddz is holomorphic in , ;ind J ^ c ' 70 FUNCTIONS OF A COMPLEX VARIABLE [OH. iv Hence ( </>(K= ( I /(*, C)d(dz-[ f I f(z, {}d{dzl JC' JG JC' I 'C -'C' Ib = bO = f f /(z, Jc ./c' 35. Derivatives of a Holomorphic Function. A function f(z) which is holomorphic in a simply-connected region enclosed by a curve C, possesses derivatives of all orders at every point interior to C. For, if z is any point interior to C, Now let A be a region which contains the point 0, and whose boundary is interior to C. Then the f unction /()/( 2) is holo- morphic with regard to both f and so long as f remains on C and z in A. Hence ( 34), Similarly, by means of repeated differentiations, it can be shewn that COROLLARY 1. If C is a circle of centre z and radius R, and M is the maximum value of \f(z) \ on C, |/M()| COROLLARY 2. If f(z) is continuous at all points of a finite ('not necessarily closed) path C, the function is holomorphic in 2 at all points which do not lie on C, and its n derivative is COROLLARY 3. If ^(a;, y)+iv(x t y) is a holomorphic function of z = x + iy, then t6(aj, y) and v(o5, 2/) have partial derivatives of all orders. 35] DERIVATIVES OF A HOLOMORPHIC FUNCTION 71 EXAMPLES IV. 1. Prove / = log , a cross-cut being taken along the negative real axis. J * z z o 2. Under the same restriction as in the previous example, prove where n may have any value except 1, and the same branch of z n is taken on both sides of the equation. 3. Prove \*e az dz = (e -!)/. f 4. Prove / cos az dz = sin (az)la. Jo 5. Shew that the residue of e az /(l + e z ) at -i is - e a7ri . 6. If K is any integer, shew that the residue of cot z at KTT is 1. 7. Shew that >\% residues of e zi /(z 2 + a' 2 ) and ze zi /(z 2 + a 2 ) at ai are e~ a /2ai and e~ a /2 respectively. 8. Shew that the sum of the residues of any rational function is zero. 9. If /(,)= A_ + _^* + ... + ,- f " , shew that the residue of f(z)l(z-x) at a is /(.') 10. If /"(2) = 2A r /(2-a) r + <(2), where ^(2) is holomorphic near a, shew ] n that the residue of /()/(*-*) at a is -2 A r /(^-a) r . 11. Shew that, if m and n are positive integers, and m<.n, M 12. Integrate ze imz /(z 4 + a 4 ), where m and a are positive, round the contour of Fig. 33, and shew that ma = ^<f^sin"!-. IQ T>^^, 13. Prove rCQ&ma; , JT . /7a TT\ / _ ^6;= - ^e V 2sm( + T Jo ^ 4 + 4 2 3 V/2 4/ 14. Integrate e iz /(z-ai), where >0, over the contour of Fig. 33, and shew that {+* a COB x + x sin x , ( L: J-a, .C 2 + (l 2 15. By integrating e iz l(z + ai\ where a>0, prove acoBx+xmnx 7 ifi p,., , ^ in ll1 -' 7 TT ~rr ma L *+& <f -'' = 2 e C08 ' where 72 FUNCTIONS OF A COMPLEX VARIABLE [CH. 17. If < a < 2, shew that /27T. + 7T\ (i) r af^dx 2?r C S \ 6 U Jo sin (**) f o lx+x* tJ3 sin^ra [Integrate - - round the contour of Fig. 37, and equate real and imaginary parts.] Prove j[ i -^_ _ -^_- 2 ^ c ?^ = ^log(l4-r), if -l<r<l f-), if r<-l or r>l. [Integrate ^ ^g^ 1 ~J!> rou nd the contour of Fig. 33, and put ^7= tan ^.] 19. If > 0, and - Tr/2 < ^ < 7r/2, prove and [Integrate s"-^- 2 round the contour consisting of the positive .?>axis, the line amp2=#, and part of an infinite circle.] 20. If ^0, prove x.x /"^(l-f.r^cosa^ , TT _y? a l) --^ = e 2 COS 2 ; /;;\ r* ^s J. 1+ [Integrate e tez /(l'+2 + 2 ) round the contour of Fig. 33.] 21. Integrate e~ z ~ round the rectangle of sides y = 0, 2/ = , x= E, and show that /+ /+ / e -(-+w> 2 ^ ^ e -W.r = N /TT. J-X J-OD Deduce: = \7r, where c is any constant. 22. Integrate e iz l(z + a\ where >0, round the square whose sides are ,r = 0, # = B, y=0, ^ = R, and shew that : ii) r <fe =rj^i ( fa. Jo ^ + Jo l + ,r-' rv] EXAMPLES IV 73 23. Integrate e~ j2 round the rectangle whose sides are .^ = 0, .v = R, y = 0, ?/ = >, where 6>0, and shew that : ' (ii) [%-* sin 2&r dx = e - 6 ' 2 / * Jo /o 24. Integrate e nz /cosh TTZ round the rectangle of sides .r= K, v/ = 0, y = and shew that r cosher 7 1 a dx = - sec = , wne re coshw.r 2 2' f2r ^Q- a -bi J/} = 27TI, if 6 > 0, 25. Prove cot c7^ .' ... 7 ' Jo 2 = - 2?ri, if 6 < 0. r2ir 26. Prove I cos"^o?^ = 0, if n is odd, and 27. Prove that, if y is the unit of circular measure : k T +x sin x , IT , r+ ao 1 - cos x I1) L T^-i 28. If a is positive, prove TT . ,,.v /"" .r sin aa; , TT =-2 Smm ' ; (u)P /o ^-^^ = 2 29. If r and 6 are positive, and < a < 2, prove [Integrate (i) zf t ~ l e ibz /(z 2 + r^) round the contour of Fig. 37, and (ii) z a ~ l e ibz /(z z -r 2 ) round this contour indented at r and -r.] 30. If -l<a<l, prove Deduce that, if -!<<!, I ^ " /"^ Jo Jo [Integrate J.A '", ^ ~ , '* round the contour of Fig. 37, and put - < >S ArtTT 74 FUNCTIONS OF A COMPLEX VARIABLE [CH. 31. Prove 7fi /1-r.V'. -1 [Integrate _ round the contour of Fig. 37.] 32. If >0, prove f^D^ Jl (1 _,-). Jo .v(x 2 + a 2 ) 2a 2V 33. If a > 0, prove _ a [Integrate -^ - ^-r round the contour of Fie:. 37.1 2(logZ- ITT/2) 34. If a > 0, prove { 1 - cos (a tan 0) } + log (cos (9) sin (a tan 6>) J_ ~ f ~ (log cos By 2 + <9 2 1 _ giaz ^ - r-r round the contou z\og(l-iz) 35. If b>0, r>Q and 0<a<2, shew that 1 _ giaz [Integrate ^ - r-r round the contour of Fig. 37.1 z\og(l-iz) [Integrate s a - 1 e ei6 7(^ 2 + ^ 2 ) round the contour of Fig 37.] 36. If 0<a<2, prove / ^_^ 2 ^ Deduce ( .... f .^- 1 -^- 1 dx A Tra / 7r6 (11) / , = log tan , / tan - x Jo log^? \-\-x 1 \ 4 / 4 where 0<6<2. 37. Let P(-s) and Q(z) be polynomials of degree m and n respectively, where m^n 2, and let Q(z) have no positive or zero real roots. By means of the integral of P(z)~Logz/Q,(z) taken round the contour of Fig. 38, prove r^Mdv- R Jo Q(^)^ where H denotes the sum of the residues of P(^)Log2;/Q(2:) (0 < amp s < 2?r) at the zeros of Q(z). 38. By integrating (Log^) 2 /(l+^ 2 ) round the contour of Fig. 38, prove floga? , I ,^0^ = 0. Jo l+.r 2 39. By integrating log(4-t)/(2 2 +l) round the contour of Fig. 33, prove Deduce iv] EXAMPLES IV 75 40. Prove / --^ dj; = 7r r - J smh.t' 4 41. If a is real, show that sinew?, TT . , aw 42. If i > - 1, w > 1, and m - n is an even positive integer, prove j" 00 sin n# sin 7i.# , _ e~' 1 e~ m J "(l + # 2 )sin# r e-tf- 1 r (logo?) 2 , 16 43. Prove / 2 ^ = QT~^S 7r - Jo 1+,P+^ 2 81v/3 [Integrate (LogzY/(I + z + z 2 ) round the contour of Fig. 38.] 44. If Kjt?<l and - TT < A < TT, shew that p jy-^dla? TT sin;? A Jo 1+2^ cos A + x- ~ sin pir sin A 45 ' [Integrate zl(u-e- iz ) round the rectangle of sides 0;= TT, y=0, y = 46. If r>0, s>0, 0<a<l, 0<6<1, a + 6>l, shew that <b ' +X dx 9 ^ a. M-- _ ~ Deduce //(cos ^) ^(fe 47. By integrating e z ' 22 /s round a suitable contour, shew that = <fo=f . x 4 Deduce / dx**^* Jo ^ 2 48. By integrating e' : /\/* along a suitable path, shew that rcos.g . _ /""sin x V/# - *J X 49. If < . < 7T/2, shew that /"-t- 00 t&n~ l xdx TTOL [Integrate log(l -^)/(5 2 -22sino.+ l) round the contour of Fig. 33.] 50. Integrate ^"/(e 2 ' 2 -!), where a is real, round the rectangle of sides = 0, .i?=R, y = 0, .y=l, indented atO and i, and shew that /" sin'/.v . 1 , /a\ 1 Jo ^ -\' f ''' = T ' [CH. V CHAPTER V. CONVERGENCE OF SERIES: TAYLOR'S AND LAURENT'S SERIES. 36.* Convergence of Series. Let S w denote the sum of the cc first n terms of the infinite series 2 w n> where the w's are real 51=1 or complex quantities ; then, if S n tends to a finite limit S as n tends to infinity, the series is said to converge or to be convergent and to Have the sum S. The necessary and sufficient condition for this is ( 23) that a number m can be found such thai, when n^im and p is any positive integer, S n +p &n i <C Or j W n+l + W n+2 + . . . 4- W n +p \ <C * Ii w n = u n + iv n , the series Stt n and ^v n converge to the real values U and V, where U + iV = S; for I ^u n ~U I i V are both less than |S n S|. Conversely, if the I series Su n and 2/y w converge to the values U and V, the series iv n ) will converge to the value U + iV, since Absolute Convergence. If the series of moduli ^ I w 71=1 convergent, the series Zw n is also convergent, since a series of this kind is said to be Absolutely Convergent. The series *Lu n and 2v n are then also absolutely convergent, since *In this and the following paragraphs some definitions and theorems on infinite series which will be found useful in the course of this work are summarised ; for fuller proofs and for further information on the subject reference may be made to Bromwich's Theory of Infinite Series. 36] THE HYPERGEOMETRIC SERIES 77 \v n \ = \Wn\- Conversely, if 2w n an d 2)v n are absolutely convergent, 2iv }l will be absolutely convergent, since .AWe. The value of an absolutely convergent series is inde- pendent of the arrangement of the terms.* Multiplication of Series. Since (u n 4- iv n )(u' m 4 iv' m ) = u n u m - v n v' m 4 iu n v' m 4 iv n u' m , the product of the two absolutely convergent series 1/w n and Sw' n is equivalent to Zw n 2u' m 2-Vn 2/v' TO 4 i S^ n Zt/ m 4- * 2v w 2u' m . Hence the product is the absolutely convergent series W^ W\ 4 (Wj -H/2 4- W z W\) 4 (tVj t(/ 3 4 ^2 ^'2 4 ^3 ^'i ) 4 . . . . Most of the series with which we shall have to deal will be absolutely convergent series. The tests for convergence of series of positive terms apply also to absolutely convergent series : the most important of these is : GO The Ratio Test. If Lim w n+l /w n \< 1, the series Vw 7l is ,l->co i* absolutely convergent: if Lim [t0 n+1 /w ll |>l j the series is divergent. If Lim iw n+1 /w n \ = l, further tests must be applied: one such /i > /> testt is the following: If IV, n+1 n n where /u, is a constant and | co n \ is less than a fixed number A for all values of ??, the series 2| w m ' is convergent if /m ]> 1 and divergent if /x ^ 1. 1. Shew that the Hypergeometric Series 1'snlutely convergent if \z\<l and is divergent if \z\>l ; while, if | s | = 1, it converges absolutely if E (y - rx - (3) > 0. *Cf. Unnnwich. 75. tCf. Bromwich, 12, 79. 78 FUNCTIONS OF A COMPLEX VARIABLE [CH. v Example 2. If R (y -.-/?)> 0, prove F(, ft y, l) = ( y ( -^_-^F(, ft y + 1, 1). Let T w denote the nth term of F(o., ft y, 1) ; then, if n = 1, 2, 3, ... , T _T ^ " 1 y/ 1.8...(y+l)(y+8)...(y+) _Yl /A T ' rp" \ y/ M+1 ~ M + 2 ' where T w ' and T" are the w th terms of F(., /3, y + 1, 1) and F(.- 1, ft y, 1) respectively. Also Hence, since LimT,, = 0, " yF(o.-l, ft y, l) = (y-F(o, ft y + 1, 1). Again, if w = l, 2, 3, ... , so that (y-o.-/?)F(a,fty, l) = (y-o.)F(cx.-],fty, 1). Hence F(o, ft y, l) = (y ,"' ) F(, ft y + 1, 1). Example 3. Shew that, if the series ^,w n is absolutely convergent, the- 00 1 series Zlog(l +w n ) is also absolutely convergent. Choose n so large that | w n \ < 1 : then Hence an m can be found such that, for n^w, where C is a constant independent of n. Therefore, if "2f | w n | < c, "z" | log (1 + w) | < Ce ; m m so that the series Zlog(l +w n ) is absolutely convergent. 37. Convergence of a Double Series. If ^ and o>. 2 are com- plex quantities such that CD.,/^ is not real, the double series + CO +00 1 SY 1 ^ is absolutely convergent. The accent indicates that the term for which m = ti = is omitted. It is convenient to assume I(ft> 2 /ft> 1 )>0 : if this is not the case, interchange w l and w 2 . 36, 37] CONVERGENCE OF A DOUBLE SERIES 79 Divide up the plane (Fig. 42) by parallel and equidistant lines into parallelograms similar and equal to parallelogram OABC, where A, B, and C are the points 2^, 2w 1 + 2o> 2 , and 2<o 2 . Since the angle AOC lies between and TT. One term of FIG. 42. the series corresponds to each angular point of the net-work, except the origin. Consider those angular points which lie on the parallelogram PQRS, the mid-points of whose sides are 2pto ly 2pw<,, where p is a positive integer. There are 2p + I points on each of the sides, and therefore, since the four vertices eacli lie on two sides, there are Sp angular points on the parallelogram. Xo\v let i/ be the shorter of the two perpendiculars from O on AB and BC. Then for each of the angular points on PQRS 1 _ 1 so that 8 wheiv tlir summation rxtnuls to all the points on PQRS. 80 FUNCTIONS OF A COMPLEX VARIABLE [CH. v Now, if the values 1, 2, 3, ... , be assigned to p in turn, all the angular points in the plane will be included. Hence ri 3< ^ 3 \r 2+ 22 + 3* +< and therefore the series is convergent. 00 38. Power Series. Let ^c n (z a) n be a power series, and 7t = let the ratio c n /c n+1 tend to R as n tends to infinity. Then from the Ratio Test it follows that the series is absolutely con- vergent within and is divergent without the circle | z a \ = R. This circle is called the Circle of Convergence and R the Radius of Convergence. Example. Shew that the radius of convergence of the geometric series + z 3 + ... is unity. At a point on the circle of convergence the series may or may not be convergent. A test for absolute convergence is given in 36. The following test is sometimes useful when the series is not absolutely convergent. Abel's Test. If the coefficients c 1 , c 2 , c 3 , ... , form a decreasing sequence of positive numbers, c n tending to zero as n tends to 00 infinity, the sum 2 c nZ n converges at all points of the unit circle i except possibly at z= 1. For, consider the series qcos + c 2 cos 20+c 8 cos 30+ . .. , < < ZTT. Let S mi _p= m+l and let s r = c so that cos cos (m + 2) = 2 s l , Then S m> P = c m+l 8 1 + c m+z (s. 2 8 1 )+ ... J f-c m+p (s p -s p . l ) Si (tfjH+l ~~ c m+->) H~ *-2( c m+-2 ~ Cm i Sp - 1 V^m+p - 1 Now s r = sin(ir0)cos { so that -I/sin J0^s r ^ I/sin J0 . (r = l, 2, 3, ...). 38] POWER SERIES 81 Therefore, since all the quantities ^m+i Cin+2> ^m+2 ^m+3> > ^m+p > are positive, S m , p = ^ m+1 ~~ nnrl Q > m+l ^^-sinp' But, by making in large enough, c m+1 can be made arbitrarily small. Therefore, since < W < TT, the series is convergent. Similarly, since the series (^sin #+c 2 sin 20+c 3 sin30+... , can be proved con- vergent if < 6 < 27T. Hence the series sn ^ converges if 0<$<27r. This theorem can be illustrated as follows : If amp z =^= TITT (n integral), the terms of the series can be (a) (b) O A 2 A 3 A, X O A, A 2 A 3 > X FIG. 43(o)(6)(c). represented by OA 1? AjA.,, A 2 A 3 , ..., {Fig. 43 (a)}, where each line makes the same angle amp z with the preceding one. These lines M.F. F 82 FUNCTIONS OF A COMPLEX VARIABLE [OH. v form a kind of spiral, and A n tends to a point, which represents the sum of the series. If amp z TT the lines will be alternately positive and negative {Fig. 43(6)} and the series will be convergent; but when amp = {Fig. 43 (c)} the method does not apply. Example. Shew that z + z 2 / < 2,+z?/3+ ... converges for \z\ = l except at 2=1 ; and deduce that the series cos W cos W COS0 + ^ + ^ + .-, a , sin W sin 3(9 , sm0 + ^ + 3 +..., are convergent if = S^TT. Multiplication of Power Series. If the two series 2^ w and 240" o o are convergent within the circle | z = R, their product c o c 'o + ( c o c 'i + c i c/ o> + ( c o c/ 2 + c i c 'i + C 2 c/ o)^ 2 + - is also convergent within that circle (cf. 36). 39. Taylor's Series. Let f(z) be holomorphic in the region bounded by a circle C of centre a and radius R, and let z be any point within C such that | z a r < R : then n\ Now, since | f ^R r for all points f on C, it follows ( 26, Cor. 7) that M where M is the maximum value of |/(f ) | on C. But this quantity can be made arbitrarily small by increasing n : hence 38, 39] TAYLOR'S SERIES 83 for all points within C. This is Cauchy's extension of Taylor's Theorem.* The convergence is absolute, for (35, Cor. 1) the modulus of each term is not greater than the modulus of the corresponding term of the absolutely convergent series Let z be the nearest singularity to a : then if z be any point within the circle of centre a and radius | z l a \ , R can be chosen so that z a <R<|0 1 a\. Thus the Taylor's Series converges absolutely at z, and therefore its radius of convergence is | z l a \: that is, the circle of convergence of the Taylor's Series is the domain of the point a. COROLLARY 1. If f(z) and its first TI 1 derivatives vanish at a, while f (n) (a) is not zero, a is a zero of f(z) of order n. For example, z = kir is a zero of sin 2 (17): this zero is a simple zero since cos*, the derivative of sin 2, is not zero at the point. COROLLARY 2. If f(z) and (j>(z), and also their first n-1 derivatives, vanish at a, while < n > Example. Prove z-*-0 COROLLARY 3. If /<*>( a ) = (71 = 0, 1, 2, ...), f(z) vanishes identically at all points in the domain of a. Example 1. Shew that, for all points within the circle | z | = 1, and deduce that j log(l+s)|^-log(l - z\). Example 2. Prove / Iog(sin7r.r)cfa;= -log 2. Integrate log (sin TT*) round the rectangle of sides .r = 0, .r=l, ^ = 0, y = R, imputed at and 1. The integrals round the small quadrants at and 1 vanish in the limit ; hence / log(sin7TA-)c?^ = il [log (sin 7T/y)- log {sin (Tr + Triy)}]dy + I log { si n (TTV jo *Cf. 43, Note. 84 FUNCTIONS OF A COMPLEX VARIABLE [OH.V Now, since w = sin ITZ = sin TTX cosh Try + i cos TTX sinh Try, as x increases from to 1, (#>0), w passes round the curve PQR (Fig. 44), FIG. 44. from P(i sinh Try} when ^ = to Q(cosh Try) when # = 1/2, and to R( - i sinh Try) when x = \ : hence amp (sin TTZ) decreases by TT, so that log (sin Triy) log { sin(7r + Triy)} = TTI. Again si Therefore Hence J log(sin irx)dx= - log 2 + J log(l - e Zwxi - 2irR )dx. But T log(l _ e 2^-2 ff B)^. < -log(l -e-^ R ), which tends to zero as R tends to infinity. Therefore P / log (sin TTX)dx= log 2. Jo Example 3. If | z \ < 1, prove (i) tan~^ = z - 03/3 + ^ 5 / 5 r (Gregory's Series) where the principal value of tan" 1 ^ is taken in each case. JGL FIG. 45. 40. Laurent's Series. Let f(z) be holomorphic in the ring- space bounded by two concentric circles C x and C 2 (Fig. 45) of 39, 40] LAURENT'S SERIES 85 centre a and radii R t and R 2 , (R 1 <R 2 ). Then if z is any point within the ring-space, so that f(z) can be expanded in a series of the form J_f ./Cf) Wc^ Now let Mj_ and M 2 be the maximum values of \f(z)\ on and C 2 ; then ( 26, Cor. 7), since \z a\=r,- ,n+l l-r/R 2 VR 2 / ^r/Rj-1 \r But these two quantities can each be made as small as we please by increasing n ; hence where A - - - 1. Since and and R 1 <|2 a -<R 2> it follows that the series is absolutely convergent for all points within the ring-space. * Cf. 43, p. 95, Note. 86 FUNCTIONS OF A COMPLEX VARIABLE [OH. v Note 2. Since f(z) is holomorphic between C^ and C 2 , the integrals round these contours can be replaced by integrals round any concentric circle C of radius R, such that R^R^Rg. It + GC follows that f(z) = 2 A^(0 a)^, where p - co Note 3. Let <j>(z; a) and ty(z\ a) represent the series 00 CO ^A p (z a) p and ^A_ P (3 d)~ p respectively. o i Then /()$(*; a)+\fs(z', a\ where 0(0; a) is holomorphic within the circle |0 a| = R 2 , and \K0; a) outside the circle |0-a| = R r Principal Part at a Pole. If the only singularity within | z a R x is at a, Rj can be made arbitrarily small. Then if n \/s(z; a) = 2-A-_ p (0 a)~ p > where n is finite, f(z) has a pole of fml order TI at a, and \/r(0 ; a) is called the Principal Part at the pole. If ifr(z m , a) is an infinite series, f(z) has an essential singularity ( 22) at a. Example 1. If f(z) is holomorphic in the region bounded by a closed curve C except at the poles a lt 2 > a > an( i ^ G r {l/(z-a r )\ is the principal part of f(z) at a r (r= 1, 2, ... , ), shew that where f is any point interior to C. [Cf. Exs. IV., 9.] Example 2. If | z \ > 1 , and the principal value of tan" 1 z is taken, shew that according as 41. Fourier Series. A uniform function F(z) which satisfies the equation F(0-fQ) = F(2;) for all values of 0, where Q is a non-zero real or complex number, is said to be a Periodic Function, and to have the period Q. It follows that, if ra is any integer, positive or negative, F(0-j- ?nf2) = F(0). If no integer 2>(j?=/=l) can be found such that Q/p is a period of F(0), Q is called a Primitive Period of the function. A function which has only one Primitive Period is said to be Simply -Periodic. 40, 41] FOURIER SERIES 87 Now let the function f(z) have the period 2&>, and let =e inz/<a . To each value of f corresponds an infinite number of values of z, differing by multiples of 2o>. Therefore to each value of f corresponds one and only one value of f(z), so that f(z) is a uniform function of f. Let A (Fig. 46) be the point 2&>, and let R denote an infinite region of the z-plane, bounded by two lines parallel to OA, FIG. 46. in which f(z) is holomorphic. Now if z is any point on a line through z l parallel to OA, z = z l + \u>, where X is real, and there- fore ^ e inz il^e iir ^, so that |f | is constant. Hence such a line is represented in the f-plane by a circle with the origin as centre, and as z increases by 2o>, passes round the circle once in the positive direction. Any portion of the region R bounded by two straight lines perpendicular to OA, and at a distance OA from each other, is therefore represented on the f-plane by a ring-space bounded by concentric circles with the origin as centre. In this ring-space f(z) is holomorphic since '*_ df(z) if'-* * ' Hence, by Laurent's Theorem, + 00 +00 where A, = C being any circle in the ring-space with the origin as centre. 88 FUNCTIONS OF A COMPLEX VARIABLE [CH. v Therefore where a p = - [ 2 "f(z)cos^dz, and 6.= - o)Jo' o> co This is Fourier's well-known expansion : it is valid for all points within the region R. The function /(z), it must be noted, is holomorphic in R. 42. Classification of Uniform Functions. Functions which are holomorphic for all finite values of z are called Integral Functions. Such functions are developable by Taylor's Series throughout the plane. From Liouville's Theorem it follows that every integral function which is not a constant must have a singularity at infinity. THEOREM 1. An Integral Function for which infinity is a pole of order n is a polynomial of degree n. For, if f(z) be such a function, then by Laurent's Theorem where 0(f) is holomorphic at f=0. Hence Therefore <j> (I/z)=f(z)- (B 1 s + B 2 z 2 + ... + B n n ). Accordingly 0(1/0) is holomorphic for all finite values of z. Hence, since 0(1/0) is holomorphic at infinity, it must, by Liouville's Theorem, be a constant, B say. Therefore f(z) = B + B^ + B 2 2 + . . . + E n z n . Polynomials are also known as Rational Integral Functions. An integral function which is not a polynomial is called a Transcendental Integral Function. The Taylor's Series contains an infinite number of terms, and thus the function has an essential singularity at infinity. Examples of such functions are e z , cos z, and sin z. An integral function f(z) which has no zeros in the finite part of the plane can be put in the form e G(z \ where G(z) is integral. For the function G (0)3= log {f(z)} has no singularities in the 41, 42] INTEGRAL AND RATIONAL FUNCTIONS 89 finite part of the plane, and is therefore an integral function : hence f(z) = e G(z) . For example, e z has no zeros except at infinity. The ratio of two polynomials is called a Rational Function. THEOREM 2. If f(z) is meromorphic throughout the plane, and if infinity is either an ordinary point or a pole, f(z) is a Rational Function. Let there be m poles 04, 2 , ..., a m , in the finite part of the plane ( 22, Th. 2, Cor.), and let the principal part of f(z) at a r be <t> r (z) = AfVC* - Or) + A ( 2 r) /(s - a,) 2 + . . . + A/(Z - a r )*v, (r=l,2, ...,m). 7)1 Then /(2)~#r(3) is finite at all finite points of the plane. Accordingly, since ^(z), < 2 ( )> > $(2)> are all zero at infinity, m must be a constant or a polynomial, say Hence /(0) = 0r(^) + V^( )' which is a Rational Function. COROLLAEY. A meromorphic function other than a Rational Function must have an essential singularity at infinity. EXAMPLES V. 1. Shew that the series : are absolutely convergent for all values of z. oo oo 2. Shew that the series ^c n z n and the series of derivatives y,nc n z n ~ l o i have the same radius of convergence. 3. Shew that the radius of convergence of the series ^nlz 11 is zero. [Such series do not define functions.] 4. Shew that the product of the series f t z n /n ! and f>"Y?i ! is 2(*+ z') n ln\ . 00 oo z n+l 5. Shew that the series S / . ^ is absolutely convergent at all points on its circle of convergence. ^ $ 90 FUNCTIONS OF A COMPLEX VARIABLE [en. 6. Shew that, for all finite values of z : (ii) cose ^l- + i...; (iii) sma =z- - + --...; (iv) cosh* = 1+^ + ^ + .... 7. Shew that, for all values of ??, the Binomial Theorem, ..., holds for all points within the circle |s| = l, that branch of (l + z) n being taken which has the value unity when z = 0. 8. If the function /(*) has an essential singularity at a, shew that l/f(z) has also an essential singularity at a. 9. Shew that the series is convergent if R (-&)>. 1/2, and find its sum. Ans. I+z. 10. Prove that, if\z\<l, T . 6:r 11. Prove l-coss 1 /.-XT- s -sin 2 1 12. Prove that, if R(s)> - 1, 13. Prove that, if | s | < 1, l{log(l+2)}2 14. Shew that the series converges if | z \ < 1, and that its sum is z/(l - z). 15. Shew that the series z z* z* z* is convergent if |s|<l and also if |s|>l, and that the respective sums are */(!-*) and !/(!-*). 16. Shew that the series 2 q^e?" converges for all finite values of z v] EXAMPLES V 91 17. Shew that, with the notation of 37, the series is absolutely convergent if A> 2. 18. Shew that the series a . 2?n + 2 2 / ...- 1 ! 2 ! % ! where w is a positive integer, is absolutely convergent if \z\ <m m /(m + l) m+1 . 19. Shew that the radius of convergence of the series 20. Shew that the series 2 ! 3 ! is convergent if | z \ < 1/4. 21. If a > 0, shew that [Integrate (**'- 1 -i2 + ^ 2 /2)/{2 3 (a 3 + 2 2 )} round the contour of Fig. 33.] cos .^ 2 + sin o^l 7 rtrt -TV / 22. Prove / Jo o x [Integrate (e**- l)/z 2 round the contour consisting of the positive x and y axes and a quadrant of an infinite circle.] 23. If a and b are positive, prove 30 24. Shew that, if a and m are positive, f a [CH. VI CHAPTER VI. UNIFORMLY CONVERGENT SERIES: INFINITE PRODUCTS. 43. Uniformly Convergent Series. Let S n (z) denote the GO sum of the first n terms of the infinite series 2 w n ( z )> whose n = l terms are functions of z ; then if, at all points of a region A, the sequence S^z), S 2 (0), S 3 (0), ..., converges uniformly (23), the series is said to be Uniformly Convergent in A. The necessary and sufficient condition for this is that, corresponding to any e,* an m can be found such that, for all points of A, where p = l, 2, 3,..., and n^m. The region A is a closed region ; i.e., the points on the boundary are included. Example. If the series 2 ^n(z) converges uniformly in a region A, and if n=l , f(z) is finite in A, shew that the series 2f(z)w n (z) converges uniformly in A. In the following three theorems it is assumed that the series oo s n W, > n (z) is uniformly convergent in the region A. THEOREM I. If Wi(z), w 2 (z), w 3 (0), ..., are continuous in A, CO the function S(z) = ^w n (z) is also continuous in A. n = l For, if z and z + kz are points of A, an m can be found such that where n^m. But, since S n (z) is continuous, an 17 can be found such that, for | Az | < 17, * It should be noted that e is independent of z. 43] UNIFORMLY CONVERGENT SERIES Hence, if | Az < >?, 93 z) - S n (z) | <. Therefore S(z) is continuous in A. CO f THEOREM 2. The series ^ 1 w n(z)dz, where C is a path i .ic j. the region A, is convergent and has the sum I S(z)dz. For, since at all points of A = f {S(z)-$ n (z)}d IJo . <el. where is the length of C. COROLLARY. If the initial and final points of C are Z Q and 0, S(z)dz, Wi(z)dz, w z (z)dz, ..., Jzo Jz Jz CO /3 are functions of z, and 2 I *p w (*)dfe converges uniformly in A, n = l Jz since a maximum value can be assigned to I. Accordingly, if a uniformly convergent series be integrated term by term, the resultant series is also uniformly convergent. THEOREM 3. If w^z), w 2 (z), w 3 (z), ..., are holomorphic in A, S(z) is holomorphic at all interior points of A, and Let f be any interior point of A, and let C be the boundary of a simply-connected portion of A of which f is an interior point. \n+ Then if, for all points of C, ^ w n (z)(z- n+] i+p wliere cZ is the shortest distance from f to C, and 7c + 1 >0. Thus 94 FUNCTIONS OF A COMPLEX VARIABLE [CH. vi )~ k ~ l dz converges to the sum J c c and therefore, since In particular, if fc Now, this integral is holomorphic (35, Corollary 2) at Accordingly S(f ) is holomorphic at and has derivatives given by Ad*u>() ,, COROLLARY. If C (Fig. 47) is the boundary of a simply- connected portion of A, and if C' is the boundary of a region A" interior to C, the series of functions of f , will be uniformly convergent in A', provided d > 0, where d is the shortest distance between C' and C. FIG. 47. Example. If the terms of the series S(z) = ^w n (z) are holomorphic in the region contained by a closed contour C, and if the series converges uniformly on C, prove that 8(2) is holomorphic within C. S a v nd uniformly convergent in the region A, provided that a con- Weierstrass's M Test. The series 2jW n (z) will be absolutely i 43,44] WEIERSTRASS'S M TEST 95 CO vergent series of positive constants ^ M can be found such that, i for all points z in A, |w n (|^M ri , (71 = !, 2, 3, ...). For, if M n+ i 4- M n+2 + . . . + M n+p < 6, . Since the moduli of the terms of the series employed in the proof of Laurent's Theorem ( 40), are Jess than the corresponding terms of the series S( r /^W^ anc ^ x](^i/^) n > the series integrated are uniformly convergent on the paths of integration. Thus the consideration of the remainder can be omitted from the proof, provided that the M Test has been previously proved. The proof of Taylor's Theorem ( 39) can then be contracted in a similar manner. 00 Kxample 1. Shew that the circle of convergence of the series ^z n /n 2 is a region of uniform convergence. 00 Example 2. Shew that the series SVC 2 " - w 2 ^ 2 ) represents a meromorphic function with poles at the points TT, 77, 877, ____ Let z be any point of the region bounded by |s| = R, where Then j * mr \ ^ mr R, where n = m + l, m + 2, ... ; and therefore 1 1 Accordingly, since the series 2)l/(ir B) 1 is convergent, 2 l/(2 2 wV 2 ) converges uniformly at all points of the region. TO Now the function 2 l/(s 2 -wV 2 ) is holomorphic at all points of the region except the poles TT, 27r, ... , WITT. Hence the given series is holomorphic in the circle except at these points. But R can be chosen so large that any assigned point lies in the circle ; therefore the series is holomorphic at all points except TT, 27r, 877, ____ 44. Power Series. Let R be the radius of convergence of the QO power series 2 c n (z a) n . Then if R^R, the area of the circle I : n = R x is a region of uniform convergence. For, corresponding to any e, an in can be found such that ^m, ! V+i- < t ( P = 1 , 2, 3, . . . ). 2 96 FUNCTIONS OF A COMPLEX VARIABLE [CH. vi Therefore, if z a =R 1? c n+l (z-a) n + 1 + c n+2 (z-a) n + 2 +...+e n+p (z-a) ^ | c n+1 1 V+ 1 + | c n+2 1 V+ 2 + . . . + c +1 , | R 1 +* Since any point within the circle of convergence can be enclosed in a region of uniform convergence bounded by | z a =R-p where | f a <R 1 <R, it follows that the series gives a holomorphic function at all interior points of the circle of convergence. + CO COROLLARY 1. If the series 2 c n(z a) n converges for CO ; a I < R 2 , it will be uniformly convergent for Example. If f(z) is defined by the series ^c n (z-a} n ^ convergent for < | z a | < R, shew that the residue of f(z) at a is c^ . COROLLARY 2. If f(z) is holomorphic and has the Laurent 00 Expansion ^ A p z p in the region R^I^I^Ra, and if infinity 00 /is the only singularity exterior to 0|=R 2 , the residue of f(z) at infinity is A_ 1 . Example 1. Prove that the residues of e l i z at the origin and at infinity are + 1 and - 1 respectively. Example 2. If n is integral and ^ 1, prove that the residue at infinity of that branch of * " which is positive when z is real and > 1 is /1.3.6...(2n-3) 1.3. 5. ..(2 -5) +( _ 1) n-i\ 12.4.6. ..(2w-2) 2.4.6.,.(2-4)T Undetermined Coefficients. Let /(z) and <(0) denote the series which converge in the region R x < | 2; a \ <^ R 2 , and let the coefficients c n , (n = 0, 1, 2, ...) be unknown. Then if </>(z)=f(z) for all points of tliis region, c n = A nj (n = 0, 1, 2, ...) ' >4, 45] UNDETERMINED COEFFICIENTS 97 For, if C is the circle | - a | = R (R 1 <R< R,), In particular, if /(,?) = for all points in the region, c n = (n = 0, 1, 2, ...). COROLLARY. If f(z) is odd, all the coefficients of even powers CO of z in the Laurent Series f(z) = ^ c n ?l are zero ; while if f(z) - r s> is even, all the coefficients of odd powers of z are zero. For, if f(z) is odd, while, if f(z) is even, Example. Consider the function l/(e z -l) : it has simple poles of residue + 1 at the points 0, %7ri, 4?, Hence, if < 1 z \ < 27r, (1) where the coefficients c , e,, c 2 , ... , are to be determined. If the sign of z be changed, Adding these two equations, we have so that c = 1/2, c 2 = c 4 = c 6 =...=0. Next, multiply both sides of equation (1) by e z 1 : then (11 --2 z 2 z 3 2j + 3 1 Hence, equating coefficients, we obtain the equations "'-2;2l + 3! = ' \\liich the coefficients c^, c 3 , c 5 , ... , can be found. 45. Additional Contour Integrals. The calculation of resi- dues by means of expansions in series is found helpful in the evaluation of many definite integrals. M . F. 98 FUNCTIONS OF A COMPLEX VARIABLE [CH. vi Example 1. Prove where m and a are real and positive. Integrate f" 2 ., over the contour of Fig. 37 ( 30). When E tends to infinity, the integral along the large semi-circle tends to zero. When r tends to zero, the integral along the small semi-circle tends to - in/a*. To calculate the residue of the integrand at ia put z=ia + : then 6 Hence the residue at f=0 is - and therefore f */ " " Jo xx^ .fa- = -W from which the required result follows. Example 2. Evaluate f* Consider that branch of is real and > 1. -, where n is a positive integer. which is real and positive when z This function is uniform in the region between the great circle C (Fig. 48) FKJ. 48. and the closed contour y consisting of small circles about 1 and 1, and the real axis between these circles. There are simple poles at +i and - i. At z = i, amp(z-l) = 37r/4 and amp(z + l) = 7r/4 : therefore 45, 46] LEGENDRE POLYNOMIALS 99 Hecce the residue at z = i is(- l)"- 1 ^ 2 ^ 2 )- Similarly the residue at z= -i is -l"- 1 ^. Thus where the integrals along C and y are taken in the directions indicated by the arrows. But ( 44, Corollary 2, Example 2) .fl,3.5...(2n-3) 1 . 3 . 5 ... (2tt-5) ,/_ iy .-i ITi r \2.4.6...(2w-2) 2. 4. 6. ..(271-4) and ^* \r c^-U W rT . 46. Legendre Polynomials. Consider that branch of in the domain of z = which has the value + 1 when = 0. Since the function has singularities at f>/(f 2 1), it can (39), for values of z such that \z\ is less than the smaller of the two quantities fv/f 2 1 1, be expanded in a series in which the coefficients are polynomials in f. The coefficient P w (f ) is called Legendre s Polynomial of order n. Shew that If we expand both sides of the equation {l_ 2( _ f)2+8 2 } -J = {i_2f(_ :uul <M|uate the corresponding coefficients, we obtain 100 FUNCTIONS OF A COMPLEX VARIABLE [CH. VI Again, from the expansion for (1 2 , it follows that where c is a small circle about the origin. Now c can be replaced by the contour of Fig. 49, described in the direction indicated by the arrows, where A and B are the ii^ FIG. 49. points \/f 2 1, C is a large circle, and y and y are small circles about A and B. The only case in which this cannot be done is when AB passes through O. But in order that this may be so, must be real and negative. Therefore, since the two quantities f+v/f 2 1 and f Vf 2 1 must be purely imaginary. Hence, by addition, it follows that f is either zero or purely imaginary. We therefore exclude the case in which f lies on the imaginary axis. The integrals along the circles C, y, and y vanish in the limit, while the integrals along DB and BD cancel each other : thus pjo-*V ds _ cos 46] LEGENDRE POLYNOMIALS AS INTEGRALS 101 where z = + \/ 2 1 cos ( The branch of Vf 2 1 considered does not matter, since cos (TT 0) = cos 0. The integrand has a singularity if /\/f 2 1 is real and numerically less than 1. In that case 2 /(f' 2 ~~-0 must be real and less than 1, and therefore f 2 is negative. Hence is purely imaginary. The imaginary axis is therefore a line of singu- larities for the integral. If f=l, P n (f) = l, so that the + sign must be taken: if f= 1, P n (f) = ( l) w , and therefore the sign must be taken. Hence, for points to the right of the imaginary axis, p n (f) = while, for points to the left of the imaginary axis, Again, in the equation P B( ) = M - _<** put 1/2 for 2 : then ~2S since the integrand is holomorphic between C and the contour- made up of y, y', and AB described twice. Thus P n (f)= 1 (V+v/r^cos 7T Jo Since P u (l) = l, we take the + sign: thus 1 TC Jo Again, let f=cos#, (0<#<7r), so that A and B become the points (Fig. 50) e iB and e~ ie . Then if, in replacing the path c 102 FUNCTIONS OF A COMPLEX VARIABLE [CH. vi by the contour of Fig. 49, the arc AB of the unit circle is taken instead of the straight line AB, we obtain Thus, if z = TT J_0 N /(2cos0-2cos0) 2 f cos( _ 2_ f ~7rJ 7rJoV(2cos0-2cos#) A In this equation let and <f> be replaced by IT and TT ; then Example 1. Prove (-l- Differentiate (1 - n(f) with regard to s : then Now multiply both sides by (1 - 2fs+2 2 ) : then But ~^ Hence equating coefficients, we have (^ + l)P n+1 (0-( Example 2. Prove , -w, 1, i-Jf\ (Cf. 36, Ex. 1.) " 1 (-!)!. 3... (2^- (l-) 2 2 , 47] EXPANSION OF cotz 103 Therefore, equating the coefficients of z'\ we have 1.1 1.2.1.2 4 !';.'> < mple 3. From Example 2 deduce 47. Expansion of cotz in a Series of Fractions. The function cot /(), where g=/=mr, has simple poles at f and ,777, (?i = 0, 1, 2, ...): the residues at these points are cotf and l/(gmr) respectively. Now consider the integral I - - dz taken round the rectangle ABCD (Fig. 51) of sides x= (w + l/2)w, y=b, where n and b are chosen so that f lies within the rectangle. To each point z on AB or BC there corresponds a point -2 on CD or DA: therefore (cot sdz ABCDA where j'-l oofesv^sdv, I.,= | JAB ^"-^- J BC 104 FUNCTIONS OF A COMPLEX VARIABLE [CH. vi On AB, z = x + ib ; therefore cot z \ = Hence 1 1, 1 ; To avoid discontinuous values of the integrand, we choose >r] then Therefore L H = l_ e -26 I 3.2 4. 7)2 _ r 2 T_ p -2i> ~rp o w / _ co t^ |^ C' / JL ^^ C/ v/ /i w -^ /T 1 ^ Lim^^O. Again, on BC, 0= (71 + 1/2)^ + ^, so that cot 01 = 1 tan i f& w Zrdy Hence I where ^ is chosen so great that (w + )& ]> r. Thus i 2 |s Therefore LimI 2 = 0. ABCDA and a~~2 ^ a definite value as ?i tends to infinity \> ( 43, Example 2). Accordingly, when n and 6 tend, to infinity, we have and therefore cot f = ~ Example 1. Shew that cosec 2 = Example 2. Integrate L _ round the contour of Fig. 51, and prove 47, 48] MITTAG-LEFFLER'S THEOREM 105 48. Mittag-Leffler's Theorem. It is possible to construct a function which shall be holomorphic except at isolated simple poles a lt a. 2 , a 3 , ... , these poles being arranged in order of ascend- ing moduli, provided that, for some integer n, the series 2 1/<V 1 ' is absolutely convergent. 00 Consider the series ^]w r (z\ where Let C be the circle z = R, where R < a p+ i \ ; then, for all points z in the region bounded by C, R Or = /u, where // = ! r and r =p + 1, p + 2, . . . . Therefore n , Hence, by Weierstrass's M Test, the series 2 U >( 2 ) converges P+I ' absolutely and uniformly in the region bounded by Q. Vj Accordingly, the series ^jW r (z) represents a function of the r = l re<[uired type in this region. But R can always be chosen so large that any assigned point lies in the region : hence the series represents a function of the required type. COROLLARY 1. If f(z) and 0(0) are two functions with simple poles of residue unity at a lt a 2 , a 3 , ... , f(z)-~ <p(z) is holomorphic at all finite points, and is therefore an integral function. Hence any function of this tyjfe-ei^be put in the form where G(z) is an integral function. COROLLARY 2. If the function is differentiated /> 1 times, a function is obtained with poles of order i> .at the points 04, ".,, ".. ..... 106 FUNCTIONS OF A COMPLEX VARIABLE [CH. vi Note. These functions have all essential singularities at in- finity, since there is an infinite number of poles exterior to every circle | z j = R, ( 22, Theorem 1, Cor. 2). 00 Example 1.' Since the series 2 V*' 2 ^ s convergent, the function is holomorphic at all points except 1, 2, 3, ... , where it has simple poles. \ Example 2. Shew that 1 \ ). [Use ^ 47.] j z -K z-mr mr + 00 Weierstrass's Zeta Function. If ^SS'V^ 8 * s the absolutely C convergent series of 37, where Q = 2mt> 1 -h2?ta 2 , then, by Mittag-Leffler's Theorem, ,. is a meromorphic function with simple poles at the angular points of the network of Fig. 42. This function is Weierstrasss Zeta Function, and is denoted by g(z), (cf. 75). The function is odd. For, if the order of summation is reversed ; i.e. if m and n are replaced by in and n ; then Hence -,)- -{1 + SS'(rV 5 Weierstrass's Elliptic Function. Differentiating the Zeta Function, we have i + ^ This is Weierstrasss Elliptic Function $>(z), (cf. 72), so th; It is holomorphic except for poles of the second order at th< points 2mfc> 1 + 2'fto) 2 , where m and n take all integral valw Since ( z) = $(%), 1p(z) is an even function. 48, 49] WEIERSTRASS'S ELLIPTIC FUNCTION 1 07 , n 49. Infinite Products. Let P n denote the product II w r , where the w's are complex quantities no one of which is zero. Then if the sequence P I} P. 2 , P 3 , ... , tends to the non-zero limit P as n tends to infinity, the infinite product ILw r is said to converge to the limit P. If P is zero, the product is defined to be divergent. If Iiw r is convergent, Lim^ n =l; for W n =*~P n [P n . l} and P n and P n _! both tend to the limit P. THEOREM I. If the series S = y) u ' is convergent, the pro- i duct Iie w will be convergent and will have the value e s . For, since the exponential function is continuous, an r\ can be found such that, if n 2>,-s e l ( Hence IIe w converges to e s . i Unconditional Convergence. If the series 2w n is absolutely convergent its value is independent of the order of summation of the terms, and therefore the value of the infinite product II'-"'" is independent of the order of the factors. When this is the case the infinite product is said to be Unconditionally 'ergent. K.i'mnple. Shew that, if the series w,, is absolutely convergent, the product 11(1 + w tl ) is unconditionally convergent. i - Example 3, 36.] 00 THEOREM II. If the terms of the series S(z) = ^w n (z) are i holomorphic in a given region, and if the series converges 00 uniformly in that region, the infinite product P(s)= H> ?r " (:) will !>' holomorphic at all interior points of the region. Kr S( : ) is holomorphic at all such points ; hence P(z) = e s(z) is .ilso holomorphic (15, p. 30), and its logarithmic derivative i^> 'ivi-n by PY?^ /-/S^ j i \ - i ' ' o v, -^ ) '^~\ / / \ \\:) = ~dz~ =: ^ Wn(Z) ' 108 FUNCTIONS OF A COMPLEX VARIABLE [CH. vi 50. Weierstrass's Theorem. It is possible to construct an integral function with zeros of the first order at the isolated points a lt a. 2 , a 3 , ..., these points being arranged in order of ascending moduli, provided that, for some integer n, the series - n i absolutely convergent. Let v( ,)_^ + I + J p+ ... + 5;, (,= 1,2,3,...). CO Then the series ^w r (z) converges (48) absolutely and uni- P+I formly in the region bounded by the circle |0| = R, where Now let (r =p + 1, p + 2, . . .), where the path of integration lies in the circle. CO Then the function ^W r (z) * s holomorphic in that region, and therefore ( 49, Theorem II) so is the infinite product II Hence the function n |Yi -~ 1 l\ <^r is holomorphic in the circle, and has simple zeros at the points ^1 > ^2 *' ^P ' Now R can be chosen so large that the circle includes any assigned point; hence the theorem holds for all finite points. Again, let f(z) be any function of the required type. Then if (f>(z) denotes the infinite product above, f(z)l<p(z) will be an integral function without zeros, and will therefore ( 42) be expressible in the form e G &, where G(0) is integral. Accordingly, the most general function of the required type is if there is a zero at the {(i - J) e^ + + --^}, 50J THE GAMBIA FUNCTION 109 Example. From Example 2, 48, we have 1 -j-ao / 1 1 \ cot*-i=Z'( --- + -). _,. \z mr inrj Hence f(cots-i)<fe= 2? [' ( -^ + } dz - 9 Jb\ z) _ Jo \z-mr mrj Iog !in = { log(l--l) + J-). 2 -oo I \ ttTT/ ttirj Therefore sin = ff' f(l - )e^l = *fl(l - -. -A _,o vA mr/ ) i V **/ Note. "VVe cannot put nn**II'(l - z/mr) : for, since the series 2{l/(2 - WTT)} 00 is not convergent, the infinite product is not convergent. The Gamma Function. We define the Gamma Function F(0) by means of the equation where y is Euler's Constant. The expression on the right-hand side of the equation is integral, and has simple zeros at the points 0, ], 2, 3, ... : thus T(z) is holomorphic except at the isolated simple poles 0, 1, 2, 3, ... , and has no zeros in the finite part of the plane, (cf. 61). The Sigma Function. The method employed in the proof of Weierstrass's Theorem, when applied to Weierstrass's Zeta Function ( 48), leads to the integral function with simple zeros at the points 2??^ + 2 ?io> 2 , where in and n take all integral values. This is Weierstrass's Sigma Function, denoted by <r(z).' By logarithmic differentiation, it follows that <T '( Z )I ( ^( Z } = ^( Z )' As in the case of the zeta function, it can be shewn that <r( z)= o-(z), so that o-(z) is odd, (cf 76). EXAMPLES VI. 1. Shew that the series 1_1_J_4.1J_ 1 ! z 1! 2 + 12! 2 + 2 3! 8+3 it presents a meromorphic function with poles at the points 0, -1, -2, -3,.... 110 FUNCTIONS OF A COMPLEX VARIABLE [CH. 00 2. Shew that the series ^z n jn\ represents a holomorphic function at all finite points of the plane, and deduce that 3. If l.Kl, prove 4. If z 1 lies within the circle of convergence of the series 2c n (z- shew that the Taylor's Series for the function at z l is Sc,/^-^)", where 5. . Prove that the residues of e z at the origin and at infinity are both zero. 6. Shew that, if A and B are the residues of e l/z z n /(l + z) at z = and z = oc : (i) 7. Shew that the residue of e 2 log ^ at infinity is (e a - 8. Shew that [Integrate e iz /(z 2 + z + l) 2 round the contour of Fig. 33.] 9. If a >0, prove f dx * Jo I 10. If > 1, prove ^ ^..-^^ 11. If a > b > 0, shew that ^-TsCaJaPIP). a + b cos 14. If a and 6 are positive, shew thafe dx 15. Shew that, if w^O, 16. Shew that vi] EXAMPLES VI 111 17. Shew that, if a and m are positive, P s * n2 .f l ' r ^' V 2 -.= . { e~ 2ma (2ma + 3) + 4wa - 3 }. 18. If - 2r < c < 2?r, prove r 30 cosh ex ^ c Jo cosh 2 7r.r ' ~27rsin(c/2) 19 Integrate gl g(l~ ?g ) round the contour of Fig. 33, and shew that (1 + 2z 2 ) 2 I x ai ' 9 .,^= I -; ' ' =^^2 1). 20. Integrate x ^ Log /, where 0<amp2<27r, round the contour of Fig. 38, and shew that / y^ <L_c? l = 7T, / j-^- ' i ^-. " a sinh 2a 23.' Prove 2 1 20 22 2s and deduce - :=- *'" 24. If ?i is a positive even integer, prove, by integrating ^ . round a suitable contour, that si H94 .97 dx e" l : ~. r == 7T" 25. Prove cosecz = - + ( -1)"Y -+ V 2-9i7T ?i7T/ 26. Construct a function f(z) which is holomorphic except at the poles s= 1, 2, 3, ..., and is such that f(s) - z cot TTZ tends to zero at each of thesepoints. Am 1 Ig 7T 27. Prove J. Shew that t&uz = - (294-l)7rJ 112 FUNCTIONS OF A COMPLEX VARIABLE [CH. vi 29. Prove that, if \s\< 1, (!+*)(! 30. Verify that : 31. Prove that, if w n = ie product IIw /? is converg 32. If \z \ < TT, shew that the product H iv n is convergent provided k = l and 2 Deduce that, if < z < TT, CH. VII, 51] CHAPTER VII. VARIOUS SUMMATIONS AND EXPANSIONS. 51. Expansions in Series by means of Residues. The theory of residues has been applied in 47 to the expansion of various functions in series of fractions. The following theorems enable us to shorten this process considerably. THEOREM 1. Let z = 'Re i0 lie on that arc of the circle js| = R for which O l = = 2 , and let zf(z\ as R tends to infinity, tend uniformly to the limit K, a constant, at all points of the arc, with the possible exception of the points for which ot e = = oc-f-e ( e arbitrarily small). Also let | zf(z) = M, where M is finite, at all points of the arc. Then Lim For (Theorem I. 30), Limff f(z)dz + R_>^ U0j and a+e f(z) dz} = i(0 - 0J K - 2ieK ; Hence Therefore Lim T 'f(z)dz = 2eM. Ja-e */(*)*].-<(4-4)K Lim f /(0)<7s = i(ft,-e i )K. 'I'lu' tlieorem also holds if there is a finite number of excep- tional values of such as oc. Integrate f(z) = e iz /z round the contour of Fig. 5^, consisting of the positive x and y axes and quadrants of the circles | z | = 11 and | z \ = r. On tlie large circle, if e^# T --:>, :/'(:) !^e^ Ksllie ; so that zf(z) tends M.F. H 114 FUNCTIONS OF A COMPLEX VARIABLE [CH. vn uniformly to the limit zero as K tends to infinity. Also, for all points on the quadrant = e --R*e< L . Therefore Hence /-/B Lim / f(z)dz = 0. R .tr, lan 2 so that, if the real and imaginary parts are equated, /"* cos x - e~ x 7 Jo .r 2 FIG. 52. Lemma. The function cot-Trs has simple poles at the points 0, 1, 2, Let these poles be surrounded by circles of radius r, where r<[l/2; then a positive quantity M can be found such that cot TTZ = M for all points exterior to these circles. For (Examples III., 11), cot TTZ =\cot\nry : hence, if y = a, (a > 0), or = , | cot TTZ \ = coth ira. Now consider the region (Fig. 53) between the rectangle of sides x= rfcl/2, y= a, (a>r), and the circle s =r. In this region,- since | cot 7r2; | = cosh ^/^/(sinVaj + sinh 2 7r?/) cot 7T0 1 = cosh 7ra/sin(7rr/ x /2) if a? = r/^/2 or = r/ v /2, and |cot7r2;| = cosh7ra/sinh(7rr/ N /2) if y = r/j2 or =r/J2. Accordingly, since unity is a period of cot TTZ, \ cot TTS \ = M, where M is the greatest of the three quantities coth Tra, cosh 7ra/sin('jrr/ A /2), cosh Tra/sinh (7ny v /2 ). for all points of the -plane exterior to the given circles. 51] LIMITING VALUES OF INTEGRALS 115 We leave the reader to prove that analogous properties hold for the functions : sec TTZ, cosec TTZ, tan TTZ ; S)TTZ 1 p"^Z I 1 COST,? sinr2; COSTS; sin rz e rz + e~ rz cos TTZ sin7T0' sin7T0' COSTTS;' e nz e~" z where TT = r = TT. v> r ^ J ' X' - 1 / 2 ^ } Y' FIG. 53. THEOREM 2. Let f(z) be a ineromorphic function, and let R x , R., , R ; , , . . . , be the radii of a series of circles with the origin as centre, no one of which passes through a pole of f(z\ and such that Lim R n = cc . Then if, as n tends to infinity, zf(z) ii ^>-'s> tends uniformly to the limit K for all points z = 'R n e ie such that 6 l = = 0. 2 , with the possible exception of a finite number of sets of points . e = = a + , and if \zf(z)\=M. for all points on the arc. f0.> Lim f(z)dz = i(0 2 -e i )K. Tin- proof of this theorem is identical with that of Theorem 1, 'xc-pt that Lim is replaced by Lim. R >ao >*> K.>'niiiplc.\. Let /(a)=cotir*/(f *) and K,, = ?i+l/2. Then, since Lim cot TTZ = - ?", Lim cot TT.S = ?', ->x, y w it follows thai Lim. ?/'(:)=/ if ^^^TT- and Lim :/(-)=- '' if 116 FUNCTIONS OF A COMPLEX VARIABLE [en. vn Also, by the Lemma above, \zf(z)\ = 'M. at all points of \z | = B H . Hence Lim and therefore, as in ^ 47, Example 2. If < r < 1 , prove & rx j * 2.^ x=+ ~~ 52. Summation of Series by means of Residues. Since the idue of TT cot TTZ at each of its poles is unity, ' where C is a contour enclosing the poles m, m-hl, ..., n, 01 cot TTZ, and no others^/^j^is meromorphic, and 2 denotes the sum of the residues of TT o8E\ir$)f(z) at the poles of f(z) within C. Similarly y f(r)={ f^ dz v'=f ^ dz 2" where 2' and Z" are the residues of respectively at the poles of f(z) within" C. + 00 1 'I Example 1. Prove 2 7 vt = "~2 * [Integrate TrcotTr^-. (x + z)~ 2 round the circle \z\ =B,, = M + l/2, and make n tend to infinity.] Example 2. If a is positive, prove V-M*-L e -7TH2/. - Va - Integrate e-^^Ke-^-l) round the rectangle of sides x- y= 1. Then, when m tends to infinity, + =0 -+i 2-*a 3 =- I ,+c 4 -- "S - 7r " 2/ ". (Examples IV. , 21 .) 51,52] SUMMATION OF SERIES 117 Example 3. Gauss's Sum. Let S M = 2 T r , where T r =2irir/H. Then r=0 T u _,.=T r ; so that S M =22 n , where 2 n stands for iTo + Ti + .-.+TV-i or ^T + T 1 + ... + T M _o+|T H according as ? is odd or even. 2 Now, 2,,-r o or 2 n -|T -T (l/2 , as the case may be, is equal to the integral of **//(** -1) taken round the rectangle ABCD (Fig. 54) of sides #=0, #=w/2, y= E, indented at O and n/2. Yyl D C -- I - 1 f O I * 1 ! *. / 2 X But and A B FIG. 54. e 2nizynd z I |-A8 e -47rBx//' w CD C 2 8 _l' < J Q l_e-2irR^ C 47^R' 2iriz2/n ^ T"/2 c 4irR.r/ /; .'ABe'- 7 "'--! " J ^"-"-1 " 4-l{ so that both of these integrals vanish when Tl tends to infinity. Again, when R tends to infinity, the sum of the integrals along the straight parts of DA tends to n'e-Zm^ln e-^iy-/" FSi^i + ^^l cos.r 2 o?^7-U sill .*-</.. Jo = ( 1 + 1) \/w/4. (^ 30, Example 5.) Similarly, the integrals along the straight parts of BC give / " ' Kinally, the integrals along the small semi-circles at O and n/2 give -^T . and - .yr,i/2 or - .^T and 0, according as n is even or odd. Hence 118 FUNCTIONS OF A COMPLEX VARIABLE [CH. vn Another Summation Formula is = ^ f -7T6 J where C is a contour enclosing the poles in, ra+1, . .., n, of cosec TTZ, and no others, and 2 denotes the sum of the residues of TT cosec (irz)f(z) at the poles of f(z) within C. Example. Shew that, if a is any non-zero real quantity, If a is small, the second series converges rapidly, while the first converges slowly. 53. Roots of Equations. The following three theorems lead up to the proof of Lagrange's Expansion. - THEOREM I. If (j>(z) is meromorphic in a simply-connected region of boundary C, then, with the notation of 31, where A<1? denotes the total increment of amp {$(2)} when z describes C positively. 17 ^< -^ 1 I (^0 ^ 1 A T r or 2*p 2*$ = ^: -. 1 , 'dz=- :ALo2Td)(0), 27T^J C 0(2;) %7Tl where A Log <p(z) is the increment of Log 0(0) when z passes round C. Hence, if <j>(z) = But A log R = 0, since log R is uniform on C ; therefore THEOREM II. Let f(z) and 0(0) be holomorphic in a simply- connected region of boundary C, and let f(z) be non-zero on C. Then, if | <i>(z)+f(z)\<\ for all points on C, f(z) and f(s) + <t>(z) will have the same number of zeros within C. For, let w= l + <t>(z)jf(z)\ then, as z describes C, w describes a closed contour in the w-plane about it'=l, not enclosing the origin, and amp w returns to its original value. Hence the increment of amp {/(0) + 0(0)} is equal to the increment of amp {f(z)} ; and therefore, by Theorem L, these two functions have the same number of zeros within C. 53,54] LAGRANGE'S EXPANSION 119 THEOREM III. If f(z) is holomorphic for < r, and is not zero at the origin, a finite quantity p can be found such that, if w ^-p, the function \^(z,w) z wf(z\ regarded as a func- tion of z, has one and only one zero in the circle \z\ = r'< r : and this zero is itself a holomorphic function of w for w \ = p. For, let p be chosen so that, if \z\ = r, ^(z, w}-^(z, 0) =>/(*) <r', provided ; i<; = p. Then (z,w)-+(z,0) ' +(*, 0). so that, by Theorem II., if w\^p, \f/(z, w) has one and only one zero, f say, within j z = r. Now, the integral _ f. -Trij . 2-Trij \[r(z, w) 27ri Z wf(z) '" taken round j 5; = /, is a holomorphic function of it; ( 34). But, if w\^p, this integral has the value f ( 31, Corollary 2). Hence f is a holomorphic function of w for | ^ | = /o. COROLLARY. If F(z) is holomorphic for \z <^r, F(f) is a holomorphic function of w for | w = p, and F(f) wliere the integral is taken round | z \ = r, ( 31). 54. Lagrange's Expansion. The results obtained in Theorem III. of the previous section can be stated thus : let f(z) and F(z) be holomorphic for \z\<r, and let f(z) be non-zero for = 0; then, if z denotes that branch of the function of w given by z = wf(z), which vanishes when w = 0, a finite region | w \ ^ p of the ^<;-plane can be found in which F(z) is hdlomorphic. The Taylor's Series for F(z) in this region can be found as follows. Let C denote the circle \z\-r\ then 120 FUNCTIONS OF A COMPLEX VARIABLE [CH. vn since \wf(z)/z\<^l, if ? 2iriJ since all the integrals in the first series have the value zero, (26, Cor. 8). In particular, if F(s) = 0, These are the well-known expansions of Lagrange. If the origin be changed to the point -f, and 0(z) be written for /(0 f ), these expansions become and where z is that root of z = g+w<f>(z) which has the value when w = 0. Example. Shew that the root of 2(1+2)"* = ^?, which is zero when v; = 0, is given by .. ,# i , . "2T 3! 41 Rodrigues Formula for P M (f). If 2; is that root of *f+w("-i)A which has the value (=/=l) when u = 0, Before differentiating this series with regard to f, we must shew that a region can be found in the -plane in which the series is uniformly convergent. 54] KODRIGUES' FORMULA 121 Let f be replaced in the expansion by \ = ik, where k is real and positive; then it follows from the theory of Lagrange's expansion that a value of p, say p-p l , can be found such that the series is absolutely convergent if \w strass's M Test, since 2 = p l . Accordingly, by Weier- provided | f \ = k, the series of equation (i) is absolutely and uni- formly convergent in w and f for | w \ = p l and for | f | = k ; and /c can always be chosen so that | f |<&. Hence (43, Th. 3) - Now where that branch of ,/(! 2w-}-w~) is taken which has the value 1 when w = ; therefore 1 + W " P - 46 > Hence, equating the coefficients of iu n in the two expansions for -^, we have Rodriguez' Formula, By differentiating the product (f 2 - l) n = (f - 1 ) M (f + l) w n times it can be shewn that the formula is also true in the exceptional cases f = 1. COROLLARY. S " 122 FUNCTIONS OF A COMPLEX VARIABLE [CH. vn Example 1 . If m i= n, I* P IH (x) P n (x) dx = 0. .'-i For let m>n : then, by repeated partial integrations, (,)p,^ = 0. Example 2. Shew that P n ' 2 (^) <&,-= 2/(2w + 1). J- As in Example 1 we have -* 1 where A ' =2 * 1 - Example 3. Shew that s= A H P n () + A H _ a P II _() + A <l _ 4 P > ,_4(s)+ where A n =2 B (w!) 2 /(2w)!. Example 4. Shew that : (i) T 3 w P n (,s)cfe=0, where ' 55. Analytical Continuation. If f(z) is holomorphic in a region S, if ^(2;) is holomorphic in a region S', which includes S and if <l>(z)=f(z) for all points of S, (/>(z) is said to give the Analytical Continuation of f(z) in the region S 7 . 00 For example, the function f(z) = ^z n is holomorphic at all o points within the circle |z| = l, the function </>(z) = l/(l z) is holomorphic except at z = 1, and 0(0) =f(z) within z\ = l. Thus <j)(z) gives the continuation of.f(z) over the rest of the plane. Example. If f(z) = ^llz tl , over what region is/(0) holomorphic, and what function gives its analytical continuation ? ^4?is. Outside |0| = 1; !/( 1). 54, 55] ANALYTICAL CONTINUATION 123 The following theorems are useful in determining the ana- lytical continuations of functions. THEOREM I. If a holomorphic function f(s) and all its deriva- tives vanish at a point a, f(z) and all its derivatives will vanish at all points in the domain of a. For f(z) = ^c n (z-a) n , where c n =f n \a)/nl, (n = Q, 1, 2,...); o thus c = e 1 = c 2 =...=0, and therefore f(z) t f(z), f'(z),..., all vanish at all points of the domain. COROLLARY. If two functions and all their derivatives are equal at a point a, and if they are both holomorphic in a circle of centre a, they are equal at all points of the circle. For the differences of the two functions and of all their deriva- tives vanish at a. THEOREM II. If f(z) and all its derivatives vanish at a point of a connected region E in which f(z) is holomorphic, f(z) will vanish at all points of E. Let A (Fig. 55) be the given point, and P any other point of E. Let a path AP in E join A and P, and let d be the shortest FIG. 55. distance from any point on AP to the nearest singularity of A*) i s that the domain of any point on AP must be at leas of radius d. On AP take successive points A, P lt P 2 , P 3 , ..., such that each lies within the domain of the preceding point. 124 FUNCTIONS OF A COMPLEX VARIABLE [OH. MI They can be selected so that, after a finite number of steps, a domain is reached which contains P. Then (Theorem I.) f(z) and all its derivatives vanish at P 1? P 2 , P 3 , ..., and therefore at P. COROLLARY. If two functions and all their derivatives are equal at a point of a connected region in which they are holo- morphic, they are equal at all points of the region. THEOREM III. If two functions f(z) and 0(0) are equal at all points of a line L in a region E in which they are both holo- morphic, the functions are equal at all points of E. For, if the points z l and z 2 lie on L, Lim = Lim Thus the first derivatives of f(z) and <j>(z) are equal at all points of L. Similarly all the other derivatives of f(z) and (j>(z) can be shewn to be equal at all points of L; and therefore the functions are equal at all points of E. This theorem is particularly important, as it enables us to extend theorems which have been proved for the real variable to complex values of the variable. For example, let and s = l: theu, if we assume that the equation sin' 2 # + cos 2 a; = 1, or f(x) = <f>(x), has been established for x real, it follows, since f(z) and 0(0) are holomorphic for all finite values of z, that /(z) = 0(z) for all finite values of z; i.e. that si Example 1. Prove P.(f)-A* Since the zeros of 1 2zf+{ 2 are i(l \/2), the expansion is valid if f!<\/2-l. Hence the- series of positive terms 2|P n (z)|R*, where R=0'4<\/2-], is convergent. But, if | z \ ^ 1, | P(*) 1 ^ | P M (t) |, ( 54, Corollary). Tims the series ^fr^-p^ 2P()f " is uniformly convergent with regard to both z and provided | z \ ^- 1, ! f | = H. 55, 56] ABEL'S THEOREM 125 Now differentiate with regard to z and in turn ; then so that ( Accordingly, if the coefficients of " are equated, P.fr)-,g-<L)-'* P r' ( '>. where |K1. Cfo CT2 But the functions on both sides of this equation are holomorphic for all values of the variable ; therefore, for all values of z, Example 2. If ze bz = w, shew that provided ?0 | < e~ l /\ b \ . [Apply Lagrange's expansion for F(z)=e az . Since the series is convergent for 'w\<T l /\b\, it follows, by the principle of analytical continuation, that the equation holds for that region.] 56. Abel's Theorem. A power series represents a continuous function at all points within its circle of convergence. If the series also converges at points on the circle of convergence, the following theorem shews that the region of continuity includes these points. XI THEOREM. If the power series ^c n p n = <f>(z) be convergent at o a point on its circle of convergence, and if z be a point within the circle, ?e n z Q n = Um<j>(z), ,. --^o ^ where z tends to Z along a radius.* Let ^ = /o(cosO-hisin 0): then c n z n = c tt p n (cos n + i sin nO) = 2 c />o naj " cos nO+i^Cnp^x" sin nO, where x = /o//o . < Hence it is only necessary to prove that if the series s^ wnere O = cco87 or is convergent, the function 2^a n x n will be continuous for O^x^l. * Cf. Brninwich, Infinite X-/Yx, 83. 126 FUNCTIONS OF A COMPLEX VARIABLE [CH. vn Now let r llt p = a H+l + a n+2 + . . . -f a n+lt . Then for any particular value of n, finite quantities H and h can be found such that H>r Mf ,,>/*, where p is any positive integer. Hence a. tl p - r n p -a^ if 0a; Similarly, a, l+1 ^ i+1 + a H+2 fc+ 2 + . . . + a B+Jl aj N +J' > //.. But an m can always be found such that | H |<e and | h forn^m; so that l^^^a^^^/.^o,^^ < Hence the series converges uniformly for O^a-^1. Conse- quently the theorem is proved. COROLLARY. If the series converges at all points of an arc of the circle of convergence, <j>(z) will be continuous on that arc. Example 1. Consider the function rendered uniform by a crosscut along the negative real axis from - 1 to - oc . V, FIG. f>(j. Let A and P be the points - 1 and z respectively, and let the circle of centre A and radius AP cut OX in E (Fig. 56). Let the integral be taken along the path ORP : then where > denotes z_OAP. 56] SUMMATION OF TRIGONOMETRICAL SERIES 127 Now, if |*1<1, and this series converges at all points of the circle |.*| = 1 except A, (Abel's Test, 38). Thus it represents the continuous function log(l+^) in the region of Fig. 56, bounded by the circle \z\ = l indented at A. Accord- ingly, if P is the point z = e ie , where -TT< #<TT, so that, if real and imaginary parts be equated, cos 20 . cos 30 3 ^ UUS^f UU t>l/ COS r- "I -... = . rO] 2 J 2 . sin 20 sin 3 ui6 2" + ~3~ ^ = in the first equation gives log 2= 1 - 1/2 + 1/3- .... The two series of 38 are uniformly convergent in the interval e^0 = 27r e, They can therefore be integrated term by term. H '-ample 2. If - TT g^ ^ zr, prove . COS26' COS 36* 0- 3. If A, B, P (Fig. 57) are the points ?', i, and z respectively shew that, for all points of the region bounded by the circle \z\ = l indented at A and B, 128 FUNCTIONS OF A COMPLEX VARIABLE [CH. vn Deduce , cos SO cos 50 f ^M, if cos is positive; (i)cos0- + _ l _...= , o, ifcos<9=0; [ - 7T/4, if cos is negative : EXAMPLES VII. 1. Integrate -{- -e~ 2 J round the contour of Fig. 52, and shew that ' 2. Prove 3. P,ove 4. Prove ] -UU* ^ 5. If - TT ^ r ^ TT, shew that : 1 2 (i) ^T e ''" + 6 ~'' 2 _ 1 1 cos r cos 2r cos 3;- ' 8 - -" z ~ 2 2 cosr cos2r cos3/ % 6. If - - < r < TT, shew that : (i) ? g r2 e~" sin r 2 sin 2r 3 sin 3r /\ TT sinrz sinr 2sin2r 3sin3r * "V-1+ 38-4 a-9 7. Shew that P/ Jo .r 2 8. If - 6 < ' < />, shew that /.% p /"" sin aA' dx TT sinh ,'o sin bx 1 +,v 2 2 sinh b cos ax x d% TT cosh a ("\ P / Jo (iv)P , /*" sin era? dx _ TT sinh a . .'o cos bx x(\ +,v 2 ) 2 cosh b ' cos ax dx TT cosh a (v)P .'o cos bxl+x* 2 cosh 6 ' ""sin ca? .rc?^; TT sinh Jr> cos fejt; 1 +# 2 2 cosh // 56j EXAMPLES VII 129 9. If m > 0, prove f* 3 dx I cos 4mx tanh x = log (coth ?/ITT). 10. If a and b are real, and TT < b < TT, prove r cosh fca: cosh(/2)cos(6/2) / r - cos ax dx = -- \^- - ^4-** Jo cosh TTX cosh a + cos 6 11. Prove 12. Shew that, if m is a positive integer, the root of z= + wz m+l which has the value when w=0 is given by + 3) . . . | ^2m + l , ^ , . . . ^ nm + l | provided | w \ < m m (m + 1)-"*- 1 1 ^|-' rt . 13. Prove that the coefficient of z tl ~ l in the expansion of {2/(e*-l)} H is ( - I)"- 1 . [Use Lagrange's Expansion for w=e z - 1.] 14. Prove that the coefficient of z n ~^ in the expansion of is 1/2. [Use Lagrange's Expansion for iv=z(2 + z)/(l + z) 2 , F(2)=log(l +z).] 15. If m and w are distinct positive integers, shew that 1, Prove 17. Prove /P-+i(*)P^(^^ 18. Prove 19. Shew that n_^^ = (2 . ? _ 1)Pn _ l(4 20. If 1 10 <l/4, shew that where w=z(l z). 21. If [ '"|</3- 1 , shew that where z is that root of log z = wz which has the value +1 when w= [In Ex. 2, 55, puta=l, b= -l,^= 22. If n is a positive integer, shew that [In Ex. 2, 55, multiply by e$~~, and equate the coefficients of 2".] M.F. I 130 FUNCTIONS OF A COMPLEX VARIABLE 23. If is real and | 1 < e~\ shew that : 1 2- *3 3 (i) cos 0=1 -0sin0 + ^0 2 cos 20 + ^03 8^30-^04 cos 40-.. [CH. [In Ex. 2, 55, put a=b=i.\ 24. Prove that, for all points within and on the circle | z \ 1, and deduce that ... cos 20 cos 30 cos 40 w 1.2 2.3 ^ 3.4 / /9\ fi = (l+cos0)logf 2 cos-J -cos 0-^sin ; .... sin 20 sin 30 sin 40 2 / n\ a = sin log (2 cos - } - sin + -5 (1 + cos 0). r Jn =log2. 25. Prove [Integrate 561 ' 26. If - 1/2 < m < 1/2, shew that f 30 sinh 2 wi# , , , ^r dx = i log sec mir. Jo x sinn x 27. Shew that, if < amp z < 27r, Deduce : round the contour of Fig. 33, and use Ex. 1 Graph the functions represented by these two series for all values of 0. 28. Shew that sin 30 sin 50 H -- ~ 29. Shew that 7T/4, if sin 0>0. _ . . /, ^ sin 0=0. -7r/4, if sin0<0. !T /9 if _! r -< #<iT 4V 2=^ = 2' vii] EXAMPLES VII 131 30. Shew that, if - Tr/3 < < Tr/3, f. COS 50 COS 16 COS 11 # _ 7r v /o~ ^5~~ ~T~ 11 = 6 31. Shew that the locus represented by I tlt'sin^sin^O consists of two orthogonal systems of straight lines dividing the (#, y) plane into squares of area 7r 2 . 32. Shew that the equation 2 I sin ttycosft#=0 represents the lines y=mir, (ra = 0, 1, 2, ...), together with a series of arcs of ellipses whose axes are of lengths TT and 7r/\/3, placed in squares of area 7r 2 . Draw a diagram of the locus. *> / _ ly*- 1 33. If f(x, y, z) = 2 i 1 sin nx sin wy sin 7?^, i=l W shew that, within the octahedron bounded by the planes xyz=Tr> 34. If shew that, for < 6 < Tr/6, r=2 cos (0 + 7T/3). Graph the curve foi 1 values of between and 27r. 35. If w = z(l +2 2 ), and the principal value of tan" 1 * is taken, prove that 4 w 3 6 . 7 w 5 8 . 9 . 10 w 7 ten-if-^-jy 3-+-^ T - ^y 7 +-.. , provided |w|<f\/3. 36. Shew that the two functions 4? tan 2 each possess only one zero within the unit circle. [CH. vni CHAPTEE VIII. GAMMA FUNCTIONS. 57. The Bernoulli Numbers. The Bernoulli Numbers B 1? B 2 , B 3 , ... , are defined by the expansion v * T^ = i - + VC IV 1 - 1 - n ^z Ln e*-} 2 + 2/ (2w)| * considered in 44. Their numerical values can be found by the method established there ; thus "i = ib ^2 = inr BS = TV> From the expansion it follows that ( l) n ~ 1 B n /(2ii)! is the residue of l/{z* n (e*-l)} at = 0. But (Theorem II, 51) where c v denotes the circle \z = (2i/+l)7r. Therefore T5 oo -I co 9 (_l)n_E!!_ = y/ L, = (_l)ny ^_ ".-' j- Example. Shew that J^+i^ + J^ + ... = |:. T/te Bernoulli Numbers as Definite Integrals. If a is positive, .+6-'' a = ?r cot OTS). <?-"<&, (52) 'C where C denotes the rectangle ABCD (Fig. 58) of sides x = 0, x=b=n+ 1/2, y = R, indented at O. The integral of (l/2i)cot (TTZ). e~ a " along the small semi-circle tends to -J, (30, Th. 2). On the contours ECDF and GABE replace (l/2i) cot (TTZ) by i i and A-f n - 57] THE BERNOULLI NUMBERS 133 respectively. The integrals arising from the terms J and + J tend to p P ^ e~ az dz and ^1 e~ az dz "JECDO "JOABE respectively. But, since e~ az is holomorphic in the rectangle, each of these integrals is equal to Thus we find where and Now so that p = J - ax dx. Y, ' 1 i t A B FIG. 58. Hence LimI 1 = 0. 134 FUNCTIONS OF A COMPLEX VARIABLE [OH. vm 2 f 6 2 Again, 1 1 5 ' LimI = 0. .-a = l_l^r2sinat/ a 2 Hence Accordingly But Hence, expanding sin at/ in powers of a (cf. Bromwich, Inf. Ser., 176, B), and equating coefficients, we have B, Example. Prove 58. The Asymptotic Expansion of Euler's Constant. Let O Til I L. \AJ% / Q C? O \ where C is the rectangle ABCD (Fig. 59) of sides o;=l, x = y = R, with small semi-circles at 1 andvjt,. D C i t FIG. 59. The integrals of (l/ / 2i)cot(7r2;).2;- 1 along the small semi-circles at z= 1 and z = n tend to J and l/(2n) respectively (30, Th. 2). On the remaining portions of C replace (I/2i)cot(7rz) by ' e -zmz_ or 58] EULER'S CONSTANT ASYMPTOTIC EXPANSION 135 according as z lies above or below the #-axis. The integrals arising from the terms ^ and + i each tend to " Thus we find = l \ l \ ( ndx i r 2y d y f R 2~ h 2tt~ h J 1 aj+Jol+^e 2 "*-! J T _ dx C n 1 dx - v 2 dx -- so that Lim 1 = 0. R-oo Hence S . But f _?SL _^_/l rgy^y - Jo 7i 2 + y 2 e^-l^?i 2 Jo e 2 ^-! ' therefore, as n tends to infinity, S n log n tends to the limit If" 2y q This limit is Eider's Constant, and is denoted by y. Thus Bo where = r 2y2k+i d y ^ i r Jo W 2 *(w 2 + y 2 ) e 2 ^- 1 < ^'+^ J 136 FUNCTIONS OF A COMPLEX VARIABLE [OH. vm 00 V* -I / r 2fc+2 Now B w = (2fe+l)(8fc+8) r 4r " r=l but lr*Slt*=^6 and lr 2 *+ 2 >l, so that CO T> Thus the infinite series 2 ( I)*' 1 srA. is divergent ^i 2kn 2k Nevertheless, if sufficiently large values of n are taken, the sum of a few (say k) terms will give the value of y to any approximation required. For R^ can be made arbitrarily small by increasing n. An expansion such as this, consisting of a finite number of terms and a remainder which can be made arbitrarily small by sufficiently increasing the variable, is called an Asymptotic Expansion. Example. If n = 10 and k = 2, shew that E fc < '000000004. 59. Convergent Integrals. In our definition of an integral we assumed that the path of integration did not pass through a singularity of the integrand /(). It is sometimes possible, however, to extend the definition to include cases in which an extremity z l of the path is a singularity of f(z). Let %' be a point on the path of integration ; then, if the ft f(z)dz tends to a definite value as tends to z lt this z (X limit is taken as the value of I f(z)dz, and the integral is said JZo to be convergent. The necessary and sufficient condition for Jz" f(z)dz should tend to zero as z and z" tend to z r The following two rules are useful for determining the con- vergency of integrals. RULE I. Let z l be a finite point ; then if a number n < 1 can be found such that (z z^) n f(z) tends to a definite limit L as z tends to z l , the integral I f(z) dz will be convergent. 59, 60] CONVERGENT INTEGRALS 137 For if z be chosen so that | (z z^) n f(z) L < e, provided |0 Zi\ = k, where k= z' z^\ t *! d/) , where P =|*-,| and this quantity can be made arbitrarily small by decreasing k. Example. Shew that the integral f~ 2 -r is convergent. -fci vUz-ZiX 2 -^)} RULE II. Let the point z l be at infinity ; then if a number n > 1 can be found such that z n f(z) tends to a definite limit L as z tends to infinity along the path of integration, the integral will be convergent. For if z' be chosen so that, for points z on the path of integra- tion between z' and infinity, z n f(z) L | < e, , where >=s, K= |L|+e and this quantity can be made arbitrarily small by increasing K. Example. Shew that the integral / e~ z z n dz, taken along a straight line making an angle < with the .r-axis, converges if ?r/2 < (f> <7r/2, and n> 1. 60. Uniformly Convergent Integrals. Consider the integral ff(z t f ) cZ0, where /(0, f ) is holomorphic with regard to both 3 c and f at all points of a region A in the z-plane which contains the curve C and at all points of a region A' in the f-plane, except for a singularity at the (upper) extremity % of C.* Let z be a point on C, and let C t be that part of C which has z' as its (upper) extremity. Then if, for all points f of A', I f(s, Qdz tends uniformly to the limit 0(f) as z tends to z lt the integi'al is said to be uniformly convergent in A r . * It is assumed that the path C is independent of f. 138 FUNCTIONS OF A COMPLEX VARIABLE [OH. vm THEOREM. If f(z, )dz is uniformly convergent in A', it is a J c holomorphic function of f at all interior points of A', and ajjsf A* f)<fc=J |^/<*. f)<fe, <=i, 2, 3, ...). Let z' be chosen so that, for all points f in A', | >/ 1 < e, where Then 0(f) is continuous in A', since \Kf) is continuous (34) and |i;|<e. Again, let f be any interior point of A 7 , and let K be the boundary of a simply-connected portion of A' of which f ' is an interior point. Then o that .+ - eL where d is the shortest distance from f ' to K, and L is the length of K. Hence I =~= f(z t ?)dz or \!s M (n converges to the limit ^ -* I rffT^i as s' tends to 2 l ; and therefore In particular, if ?i = 0, /(^, f')^ converges to the limit Jo , S o that 0(r) = - - Now this integral ^TT'UK ^ is holomorphic (35, Corollary 2) at f. Hence 0(f) is holo- morphic at f ', and has the derivatives Example 1. Integration under the Integral Sign. If C' is any path in A', f <(0^= f ( /(*, Jc'* ~'c J<y 60] DIFFERENTIATION UNDER INTEGRAL SIGN 139 For ( <t>(Qd=l VKCK+f rjdf Jc' Jcr JGf = 11 fa K<fe+ f V<%, (34, example). JCi J0 Jc' Now / 7?d <eL, where L is the length of C'. Hence / / /(z, {)ddz \ Jc' J<h JO tends to the limit / </>(O^C ' so that -'C' f <KK= f f /(*, Mi**. Jc 1 Jc Jc' The following two rules, the proofs of which are similar to those of 59, are useful for determining the uniform convergency of integrals. RULE I. Let the extremity z l be a finite point ; then if, for all values of in A', a number n < 1 can be found such that (0 z^ n f(z, f) tends uniformly to the limit L(f) as z tends to z lt the integral I f(z, )dz will be uniformly convergent. Jc RULE II. Let the extremity considered be at infinity; then if, for all values of f in A', a number n^> I can be found such that z n f(z, f) tends uniformly to the limit L(f) as z tends to infinity along C, the integral I f(z, g)dz will be uniformly convergent. Example 2. Consider the integral 4>(z)= / e^^dt^ where R(2)>0. Jo Let a ^ x= R(2) ^ 6, where a > ; then, if * > 1, | re-*?' 1 1 = e-$* +n - 1 ^ e- f t b+n - 1 . But Lime~ i! ; 6+n - 1 = ; hence the integral converges uniformly at its t >-QO upper limit. Again, if t < 1, | t n e-*t z - 1 \ ^ e~'r +n - 1 . Now choose a and ?* so that a < I and (1 - a) < < 1. Then hence the integral converges uniformly at its lower limit. Now, if R( 2 )>0, a and b can be chosen so that a^~R(z)^b ; hence isholomorphicfor R( 2 )>0 and has the derivative f e^^ Jo It is easy to verify, by partial integration, that: (i) (ii) </>(!)=! ; (iii) if m is a positive integer, <j>(m 140 FUNCTIONS OF A COMPLEX VARIABLE [CH. vm Example 3. Consider the integral taken along a straight line which makes an angle ^ with the .r-axis. If satisfies the inequalities where r is positive, all the values of f are excluded which make 2 +2 2 =0. Now, if \z\=p>Il, fel'g* <"^ which tends to zero as p tends to infinity if l<w<2 ; hence the integral converges uniformly at its upper limit. Again, if \z\ = p<r, which tends to zero as p tends to zero if 1 > n > ; hence the integral converges uniformly at its lower limit. Accordingly, <(f) is holomorphic, provided ^r ir/2 <ampf<^r + 7r/2, Example 4. Consider the integral taken along the path of Example 2, where - 7r Let f be confined to the region defined by Then, if || = / ir/2. where x ( = /acos^) tends to infinity with />. Now, if n< 2, hence the integral converges uniformly at its upper limit. Again, log ( 1 _^ 27r? ) = - log z + log/(s), where /(0) is holomorphic at 2=0. Therefore, by Example 2, the integral converges uniformly at its lower limit. Accordingly, <() is holomorphic, provided 60, 61] THE GAMMA FUNCTION 141 61. The Gamma Function. Gauss's Definition. Let T(z) denote the function n ! n z Then so that this definition is equivalent to that of 50. The following properties can easily be deduced from the latter definition : (ii) r(m + l) = m!, where in is a positive integer; (iii) T(z)T(l-z) = -^; SmiTZ (iv) the .residue at z = in, where in is zero or a positive integer, is ( l) m /m!. The Function ^r(z). Similarly, if \[s(z) = -=- log T(z+l ), we have: dz (ii) ,/,((>)= -y; (iii) ^() = l+l + ...+l (iv) \fs( z l) = \ls(z) + 7r Gauss's Function II (z). The notation 11(2;) is frequently used instead of T(z + 1) : thus U(z) = zli(z-l\ n(m) = m!, and U(z-l)TL(-z) = Tr/sm TTZ. Euler's Definition. The Gamma Function may also be defined as the integral I e'^ 3 ' 1 ^, provided R(2;)>-0. We shall now Jo prove that the two definitions agree for values of z which satisfy this condition. 142 FUNCTIONS OF A COMPLEX VARIABLE [CH. vm If R(z)>0, and n is a positive integer, then, by partial integration, + Thus, writing y u/n, we have so that r(2;) Now let /(t*) = 1 - & (l - -Y, \ lu/ where Q^-u^n', then Thus /(^) is an increasing function ; so that I- >Q, or 6-^(1--) . n/ ~ Again, fu fw / ^,\n-l^, x,w fu f(v)dv=\ ^(l--) -^<- o Jo \ n/ n nJo Accordingly, if ^ u ^ n, (fii\n n,2 1-) <1L. W ~2n Now we can write vdv e u -^~ o f'' f'' / ii\n - e-"^- 1 ^- [l-.-W- 1 ^ o \ Ja J\ n/ Let a be chosen so large that, for all values of n greater than a, j n e- u u z - a Cn / n, \ ?i and therefore ( 1 -- ) u z ~ 1 du J \ n/ < e ; 61] then EULER'S DEFINITION 143 f/t o n Hence, if R(V)>0, r(z) = Limf'Yl--Y l -*J<A n/ Gamma Function expressed as a Contour Integral. Euler's expression for F(0) can be replaced by the following, which holds for all values of z. Consider the integral I e~ Jc t where C is the contour of Fig. 60, with its initial and final points at infinity on the posi FIG. 60. FIG. 61. tive -axis. The initial and final values of amp f are taken to be and 2?r respectively. Now replace C by the contour of Fig. 61, consisting of the -axis from + oc to e, the circle | f | = e, and the -axis from e to +x . Then, if R(V)>0, we have, when e tends to zero, Now the functions on both sides of this equation are holo- morphic for all values of z. Hence the equation holds for all values of 0, and 1. Prove that _ 1 () where C denotes ;i ]);itli which starts from - oo on the J-axis, passes round the origin in the positive direction, and ends at - oo on the -axis. The initial and final values of amp are taken to be -TT and TT respectively. 144 FUNCTIONS OF A COMPLEX VARIABLE [CH. vm Example 2. Gauss's Theorem. IfR(y-.-/3)>0 For ( 36, Example 2) F(oc, ft y, 1) xF(a, ft y + rc+1, 1). C = Lim -~ and Lim F(o., ft y + w + 1 , 1) = 1. Example 3. If E (y ) > 0, R (y - a. - /?) > 0, shew that F(--, -ft y-a.-ft l) = F(oc,fty, 1). 62. The Beta Function. Consider the integral of f(z) = zP-\l-z)<i-i taken round a closed contour which starts from a point A (Fig. 62) on the #-axis between and 1, and is composed of : FIG. 62. (i) a circuit APA round z = 1 in the positive direction ; (ii) a circuit AQ A round z = in the positive direction ; (iii) a circuit ARA round z = 1 in the negative direction ; (iv) a circuit ASA round z = in the negative direction. After describing this contour /( z) returns to A with its initial amplitude, which we assume to be zero. The integrand is a multiform function ; but since, at every point of the path, the branch integrated is uniform and continuous, the definition of 26 holds for the integral. The notation rd+.o+.i-.o-) f(z)dz is used to denote this integral. 61, 62] THE BETA FUNCTION 145 The path APA can be deformed into the contour consisting of : the ic-axis from A to 1 e, the small circle z 1 1 = e described positively, and the #-axis from 1 e to A. Such a contour is called a (positive) Loop. If it had been described in the opposite direction, the loop would have been negative. Similarly the circuit AQA can be replaced by a positive loop about the origin, and the circuits AR A and ASA by negative loops about z 1 and z = () respectively. As z describes the circular part of the first loop, the value of f(z) changes from f(x) to f(x)e 2qiri ; similarly, the descriptions of the circular parts of the other three loops give/(z) the values J-(x)e 2( ^ + ^ iri ) f(x) e* piri , and f(x) respectively. We now make the radii of the circular parts of the loops tend to zero ; then, if p and q are real and positive, f(l+, 0+.1-.0-) *-i(l-.*)r-i<Zs J p * 1 ***} xP- l (l -x)*- 1 dx Jo , q) Now the functions on both sides of this equation are holo- morphic in p and q ; hence the relation holds for all values of p and q. Accordingly, if we define ~B(p, q) by the equation we have, for all values of p and q, m+,o+, i-,o-) Example 1. With the same initial conditions, shew that /(!+, 0-.1-.0+) Example 2. By means of the transformation #=(2-l) 2 , shew that, if R(p)>0, Deduce that, for all values of p, (i) The latter equation gives the Duplication Foi-mula for the Gamma Function. I.F. 146 FUNCTIONS OF A COMPLEX VARIABLE [CH. vm 63. The Asymptotic Expansion of the Gamma Function. From the expression we derive the equation Now let C be a closed contour (Fig. 63) consisting of a semi- FIG. 63. circle of radius p + 1/2, where p is an integer, part of the ^/-axis, and a small semi-circle at O ; then, if R(2;)>0, V 1 J_ V(^ + ^) 2 ~2 7 ri The integral of (l/2i) cot (TT^ ) . (z+)~ 2 round the small semi- circle tends to l/(2z 2 ), ( 30, Th. 2). On the remaining part of C replace (l/2i) cot (xf ) by or according as I(f)<0. The integrals arising from the terms I and -f \ each tend to 63] ASYMPTOTIC EXPANSION OF GAMMA FUNCTION 147 Thus we find V l 1 W where Lim 1 = 0. Thus Hence (60, Ex. 1), where the constant K must be real, since all the other terms are real when z is real. Therefore where K x is a real constant, and (60, Ex. 4) Now, since T(x so that K=-(a5-f-i)logl + - + l+ J(aj)-J(aj hence LimJ(oj) = 0, and therefore K = 0. X >-co Again, since T(z)F(lz) = Tr/simrz, TQ I iiQiy iu)= ^ e ~ Therefore where tan~ 1 (2it) denotes the acute angle whose tangent is 2u. - f log(l-e-2m ? )^ 77= _^_r i iog(i_ a; )^ j where x = e-27rr, 118 FUNCTIONS OF A COMPLEX VARIABLE [OH. vm Now, if x and y are positive, xdrj -vX^ + rf- Thus LimRJ(J + m) = 0; so that (40, Ex. 2), K' = logv/27r. Therefore Again, let J^(z) denote the integral ( 60, Example 4), taken along a straight line making an angle \]s with the -axis, where 7r/2 < i/r < 7r/2 and z=f=Q. Then, since, for values of amp f between and \/r, tends uniformly to zero as f tends to infinity, J^(a?) = J(cc (30, Th. 1). Now J^(0) is holomorphic for the region R^ defined by Also, corresponding to any point z for which TT < amp z < ?r YA FIG. 64. and z=f=Q, & value of i/r can be found (Fig. 64) such that the positive aj-axis and the point z both lie in R^,. Accordingly, by 63] CONSIDERATION OF THE REMAINDER 149 the principle of analytical continuation, since T(z) and Iog0 are holomorphic provided ?r < amp z < TT and z=f=Q t = log \/27T + (Z %) log where J. W - But,(30,Th. 1), Hence Also the least value of |dfif | is the perpendicular distance of z from the line uQvJ (Fig. 64). Thus, zi\^\\z\, where X = cos ((j) i/r), (amp z = <p)', so that < X = 1 ; hence 1 Therefore 1 j^-L^ldfl, (39,Ex.l) B n+ , The infinite series B 1 B 1 is divergent (cf. 58) ; but J n (z) can be made arbitrarily small by increasing z, so that, for sufficiently large values of z\, 150 FUNCTIONS OF A COMPLEX VARIABLE [OH. vni a finite number of terms gives the value of the function to any approximation required, provided ?r < amp z < TT. The series is therefore asymptotic. COROLLARY 1. For all values of amp z such that - 1/2 e- z ) tends uniformly to the limit unity as z tends to infinity. COROLLARY 2. If z is real and positive, we can take i/r = 0. Then X = l, and T> T I T (z\ I ^ " *W I ^ so that the remainder after any term in the series for logT(z) is numerically less than the succeeding term. COROLLARY 3. m! = \/27rra( j e 12m , where < 6 < 1 : this is _ /771\ m known as Stirling's Formula. The expression \f2jnni J is usually spoken of as the approximation to m! when m is large. Example I. Prove tan- 1 1 _ ^ =\(\ - where the principal value of tan" 1 ^ is taken. Let C denote the rectangle of Fig. 59 ; then log(l .2.3...w)=T- . ATTl c Hence, by the process employed in 58, we obtain Again, log(l . 2.3 ... so that Hence, if ?i tends to infinity, we have fStM Example 2. She w^ that, if - ?r < amp z<ir, ,-a =1. 63] THE HYPERGEOMETRIC FUNCTION 151 Example 3. Shew that, if - ?r/2 < amp < Tr/2 and f =f 0, j a+xi ^ r a+Xi {* TTdz_ _ 2irtV_ fl _., U ^ 2iriJ- a -i r(z+l)iin~~' ' where a> and the path of integration is a straight line. If z=~ReM, where -7r<0<7r, Lim 2->00 ! 1 = Lim i -r,. . = Lim 1 IZ^zW* :Lim . g R cos 0(1- log R)+R sin 0.0 Hence, if = pe^ and 6 + 0, =v / 2^Lim A- gBcos(l+logp-logR)+Rsin(-*Tir) ^ Lim ,_>*> according as sin is positive or negative. Accordingly, if 7T/2 ^ 6 ^ e, or if - Tr/2 ^ (9 ^ - e, or if - a ^ R cos ^ ^ 0, tends uniformly to zero as z tends to infinity. Thus (60, Rule II.) the given integral is uniformly convergent. Next, if -e^ifl^e, let z=H m e i0 , where R m =m + l/2 and ra is an integer ; then Lim - + 1 ) Sin 7T2 where 2M ^ ! cosec 772; ! ( 51, Lemma). Hence tends uniformly to zero as m tends to infinity. It follows ( 30, Th. I.) that the given contour can be replaced by a closed contour consisting of the line x -a and that part of the circle |s| = where m may be increased indefinitely, which lies to the right of this line. Now the only poles within this contour are those of 1 /sin TTZ ; hence TT^Z _ _ i * _ ~ + " Example 4.* Shew that the integral 2^'J where -TT< amp(- ^)< TT, and the integral is taken upwards along a straight line (Fig. 65) parallel to the y-axis, with loops, if necessary, to ensure that the poles 0, 1, 2, 3, ... , are to the right of the contour, while the poles -a., -0.-1, -0.-2, ..., -/?, -/?-!, -/3-2, ..., are to the left of the contour, is uniformly convergent. Negative integral and zero values *Cf. E. W. Barnes, Proc. Land. Math. Soc. t Ser. 2, Vol. 6, Parts 2 and 3, 152 FUNCTIONS OF A COMPLEX VARIABLE [CH. vm of <x and ft are excluded, since the curve could not, under such conditions, be drawn. Also shew that, if | f < 1, the integral has the value while, if | f | > 1, it is equal to -a * -a-3 -a-2 -jS-2 -/3-1 -j8 X - yj Y FIG. 65. Firstly, let - =/>e^, where p< 1 ; also, let z=~Re iQ . Then, if e ^ O^TT/^ or if - 7T/2 ^ ^ ^ - e, where tan e < i log (1/p), Lim according as sin 6 is positive or negative. Accordingly, since TT T( -*)<-) or -^ r(y+z) tends uniformly to zero as z tends to infinity. Thus the given integral is uniformly convergent. Again, if - e ^ 6 ^ e, so that, if R = m-f 1/2, where m is integral, 63] ANALYTICAL CONTINUATION 153 tends uniformly to zero as in tends to infinity. Hence it follows, as in Example 3, that the integral has the value oL o r(y) Secondly, let p > 1 . Then, since T(l~y-z)T(-z) V *' -- L-2l --* sn 7r it can be shewn as before that the integral is uniformly convergent, and that the path can be replaced by a closed contour consisting of the given line and an infinite semi-circle to the left of the ?/-axis. The required expression for the integral is then obtained by taking the sum of the residues within this contour. An exceptional case occurs when a. and j3 are zero or differ by a positive integer. Let a.=/3 + 7n, where m is zero or a positive integer; then the integrand has poles of the second order at the points -., -oc-1, -0.-2, ____ Now the integrand can be written so that the residue at the point a. n is / iv n* _ r( ' Hence the integral is equal to 003 + 1). .. .--. Let amp = i/r ; then, since - TT < amp ( - ) < TT, it follows that, if < \ff ^ TT, mp(-^)=i/r-7r ; while, if > ^ = -TT, amp(-^)=i/r + 7r. Accordingly, the analytical continuation of F(a, j8, y, ) when | f | > 1 is according as 0<amp^7r or -7r^amp^<0. If OL and ^8 differ by an integer, the corresponding changes must be made in the expression. If a cross-cut is taken along the real axis from 1 to + oo , the function is then uniform in the whole -plane. Example 5. Prove j* + \og{r(z)}d z =]og^+z \ogz-z. If x is real and positive, , where 0<6'<1. 154 FUNCTIONS OF A COMPLEX VARIABLE [OH. vm Hence * +l \og{T(x)}dx where = log -s/277 + x log x - x + t(x\ where t(x) tends to zero as x tends to infinity. Again, / log { F (x} }dx = log x ; (jL$Gj& so that J log { F (x) } dx = K + x log # - #, where K is a constant. Thus K must be log >/27r, and therefore / 1 og { F (#)}<&;= log \/27T + .# log #-#. J* Now the functions on both sides of this equation are holomorphic for all values of the variable, provided that a cross-cut is taken along the negative real axis from to - x . Hence (55) fZ+l I I log{F()}c?.s=log\/27r + zlog2 z. h EXAMPLES VIII. 1. Shew that ^4-n...^4.9.-n __ = 2 2 - 1 . 1.3. 5. ..(2/i- 2. If 2a=26 (Examples VI. 31), shew that Where 3. Prove that 4. Prove that 5. If m is an integer, shew that 6. If a and 6 are real and >0, and K ()>() shew that (i\ 1_ / e i(t>+xi) MX (ii) f e-*^**,. .^1 = 0. 63] EXAMPLES VIII 155 [In (i) integrate e az z~ n ~ l along x= b, and shew that this path can be deformed into that of Example 1, 61 ; in (ii) integrate e-^-"- 1 round the contour consisting of x=b and an infinite semi-circle.] 7. If p is a positive integer and E(w)> - lj shew that ~ [+> z?(z 2 - I) n dz=2i sin (^ where is the initial point, and the initial value of amp(2 2 -l) is TT. Deduce that, for all values of n, the integral vanishes when p is odd, and that its value when p is even is 8. . Shew that, for all values of z, 10. If ft is a positive integer, shew that 11. If R(2)>0 and amps=\/r, shew that Deduce that the expansion is asymptotic if -7r/2 < ^<7r/2. [Replace the path of integration by a straight line from O to infinity which makes an angle ^ with the positive real axis, and shew that, for points on this line, t 2 + l \ i^cos 2 ^.] 12. If - 7T/2 < amp z=yjs< Tr/2, prove the asymptotic expansion f Jo n-i ..... - 1 where | R n | 13. If s< 1 and -?r/2< amp2=^<7r/2, prove the asymptotic expansion where | B. | < 14. If E(v-oL- / 8)>0, prove Bin ra r ^ \. - y + a, 1 - g + a. 1) 156 FUNCTIONS OF A COMPLEX VARIABLE [OH. 15. If R(y-o.-/3)>0, prove A y, l)-co8,ra^ffiF(a, 1 - 16. If R(gO>l, prove 17. If RQo - a) > 0, prove EQo-a, g) 18. If R(p + s) > 0, prove ' s(s-l) 19. If = a + ^y, where 5 is a positive constant, shew that the limiting value of | F(l +2) | when y tends to + oo is 20. Shew that the analytical continuation of F(o., /?, y, 1/z) for | z \ < I is according as < amp Z^TT or w ~ amp z < 0. 21. Shew that where the integral is taken along a contour similar to that of Example 4, 63. Values of OL, j8, y, 8, which would make it impossible to draw the contour, are excluded. 22. If RQt?)>0, shew that pT(n) 23. Prove r(z) = ~Limn z B(z, n). n *oo 24. If R(s)>0, shew that 25. Shew that, if R(s)> - 1, and deduce that y = vm] EXAMPLES VIII 157 26. Shew that, if R(z) > 0, R(f ) > 0, Deduce f e-' Jo 27. Shew that (iii) 28. Shew that r(a)ro8) 29. If R(y-o.-j8)>0, prove 30. If a. and /3 are real, shew that f r(<x) Infill | l|r(oL + l/?)|J nlo I 31. Shew that, if z+ -1, -2, -3, ..., deduce that ^(^) + 7 = 2 - 2 log 2. e- u t z - l dt is holomorphic in , and that r(e)-e%' T e-*f~*dt Jo [Integrate e-$ z ~ l round the contour of Fig. 52, 51, and apply the inequality r - >- to the circular part of the contour.] u TT 33. If < R(2) < 1, shew that 34. If < R(s) < 1, shew that fcos ^ . ^^ = T(z) cos 35. If - 1< R(z) < 1, shew that jTsin . ^rf<= F(a) sn 36. If < R(s) < 1, shew that 158 FUNCTIONS OF A COMPLEX VARIABLE [OH. 37. If < R(s) < 2, shew that 'sint , 38. If B() > 0, shew that (i) i= (iii) i = (_ 39. If r > 0, - 7T/2 < $ < 7T/2, shew that (i) (ii) 0= o t 40. If - TT < < TT, shew that t (i) log (ii) = [Put 2 = 1 +e^ e in Example 38, (ii).] 41. 42. If R() >- 1, shew that r p. I _ /z LI ir?* l- Also, if M is the maximum value of for 0^ t^ 1, /o 1-* Now make ra tend to infinity, and use Example 31.] 43. If 27^0, -1, -2, ..., prove deduce that ^( z ) _ i og g = f (l og ( 1 + J_ ) I ) . n =o I \ z + nJ z + n + 1) 44. If E(g) > - 1, shew that vm] EXAMPLES VIII 159 t_ e -tz\ e -tz_ e -t(z+l)\ / e -t(z+n)_ e -t(z+n+l)\ _ -t(z+l) _ -. (z+2) ___ - Now make n tend to infinity, and use Example 43.] 45. Prove ,= 46. If < R(s) < 1, prove ri /- i _ # 2 ^= H^F^J Jo 1 ^ fr 47. If E(^)>-1, prove _ o/ 1/2 (ii) deduce .Apply the transformation l-f. = e T to the third of these integrals, and use Ex. 44.] 48. If H(z) > 0, shew that 49. If R(^) > - 1, R(z 2 ) > - 1, R^ + 2. 2 ) > - 1, shew that 50. If shew that [CH. IX CHAPTER IX. INTEGRALS OF MEROMORPHIC AND MULTIFORM FUNCTIONS : ELLIPTIC INTEGRALS. 64. Integrals of Meromorphic Functions. If f(z) is holo- morphic in a simply-connected region C, F(V)=| f(z)dz is Jz liolom orphic in that region, provided that the path of integration lies entirely within C. If, however, the region C contains one or more poles of f(z), the value of F(z) will not necessarily be independent of the path of integration, and F(z) may be a multi- form function. Each branch of F(Y) will be holomorphic in a simply-connected region containing no singularity of f(z). The path of integration, of course, must not pass through a singularity For example, consider the integral 1 z~ l dz taken along the FIG. 66. path C of Fig. 66 from 1 to z. This path can be replaced by a positive loop from 1 round O and the straight line L from 1 to z. 64, 65] THE LOGARITHMIC FUNCTION 161 The integrals along the straight parts of the loop cancel, while the circular part gives the value 2-Tri ; hence f dz f dz -=| Jc s JL z c L Now any path from 1 to z can be replaced by a number of positive or negative loops from 1 about O and the line L. Hence the most general value of Log z I z" l dz is J -f 271? = log z + 2mri, L 2 where ^ is an integer. This agrees with the results of 18. Similarly, if a uniform function f(z) has poles a 1? a 2 , ..., of residues R lt R 2 , ... , in C, the path from to z can be replaced by a series of loops from z about a lt a z , ... , and a straight line L from to z. The most general value of the integral will then be where 77^, m 2 , ... , are integers. If, however, the residue at the pole is zero, the integral round the corresponding loop is zero, so that the integral is uniform in the domain of the pole. Thus r- = z~ l is a meromorphic function throughout the plane. Example. Verify, by integrating round suitable loops, that where m is an integer. 65. Integrals of Multiform Functions. If the path of inte- gration of a multiform function f(z) does not pass through any singularity of f(z), f(z) will vary continuously along the path, and the definition of 26 still holds for the integral. As in the previous section, the values of F(z)=l f(z)dz may differ with J-o the path ; and the path can be replaced by a series of loops about the singular points, followed by a straight line from z to z. ]. Let F(s)= / z~ l! -dz, where the initial value of - 1/2 is unity ; the integrand has branch-points at the origin and infinity. M.K. 162 FUNCTIONS OF A COMPLEX VARIABLE [CH. ix The loop about 2=00 consists of the line AB (Fig. 67), where A and B are the points z = \ and 2 = R (E large) respectively, the circle BCD or U| = R FIG. 67. described negatively, and the line BA. But this path can be deformed into a negative loop from A round O. Hence we need only consider the effect of the loops about O. Let L denote the positive loop from 1 about O ; then, since Lim z X 2~ 1/2 =0, 2->0 the integral round the circular part of L tends to zero with the radius ( 30, Th. II.). Also, as z describes the circle, amp z increases by 2?r; so that amp2~ 1/2 decreases by TT. Thus z~ l/2 changes from 1/V.r'to 1/V# ; hence c dz _ ro dx ri dx . JL Jz }\ \/# Jo ^x A description of L- 1 , by which we denote the loop L described negatively, gives the same result. Since z~ lf2 returns to A with the value -1, a second description of L or L" 1 will give the value 4, and bring z~ l/2 back to A with the value + 1. Thus an even number of loops gives the value 0, and brings z~ l/2 back to A with the value + 1 ; while an odd number of loops gives the value - 4, and brings z~ l/2 back to A with the value - 1. Hence the general value ofF(z)is 2 where w denotes the integral / z~ 1!2 dz along a straight line from A to z, with + 1 as the initial value of 0~ 1/2 . Example**. LetP()= )<&,where/(*)= W(l -2 2 )and/(*)=l initially. Also let A and B denote positive loops round the branch points + 1 and - 1 respectively. Since Lim (z-l)-jr- ^ = 0, z->i vU-^v the value of the integral round A or A" 1 is C, where 65, 66] THE INVERSE SINE FUNCTION 163 and f(z) returns to O with the value 1. Two successive integrals round A or A" 1 give the value zero, and bring f(z) back to O with the value +1. Similarly B or B" 1 gives the integral C, and two successive descriptions give the integral zero. Successive descriptions of A and B or of B and A give 20 or 2C, while f(z) regains its initial value +1 at O. Accordingly, if w denotes / f(z)dz taken along a straight line from O to 2, with initial value +1, the general value of ~F(z) is mC + (-l) m w, where m is an integer. To evaluate C we proceed as follows : make f(z) uniform by a cross-cut from - 1 to + 1, and choose the branch of f(z) which has the value +1 at the origin on the lower side of the cross-cut. Then, at a point on the #-axis to the right of 2=1, amp^/(l 2 2 )^=7r/2 ; so that /w- > where *J(j? 1) is positive. so that //()<&, taken positively round an infinite circle, has the value 2;r. But the great circle can be deformed into the loops A and B taken suc- cessively, and the value of the integral is then 2C ; hence C = TT. Thus the general value of sin" 1 z is given by = ( T It follows that the inverse function 2 = sin w has the property sm{w7r + ( l) m w}=sinw. rZ Again, since / f(z)dz= -w, it follows that z=ain( w). But z = si thus sin( w>)= sin w, so that sinw is an odd function. Many of the other properties of the sine function could also be deduced from those of the integral / dz J ON /(l-2 2 )' 66. Legendre's First Normal Elliptic Integral. Let = f(z)dz, o where /(z)= {(1 z 2 )(l k*z z )} 7 , and k is a positive proper fraction. The initial value of f(z) at z = is taken to be +1. The integrand has four branch-points, +1, 1, +l/& v -^lfc The loop A from O about 1 gives the integral 2K, where Ii fl x o S /{(1 -a 2 )(l-fe 2 )}' and f^ returns to with the value 1. Two successive integrations round A give the value 0, and 164 FUNCTIONS OF A COMPLEX VARIABLE [en. ix bring f(z) back to O with the value 1. Similarly the loop B about 1 gives the integral 2K, and two successive integrations round B give the value 0. Successive integrations round A and B or round B and A give the values 4K and 4K respectively, and f(z) regains the value + 1 at O. Since a straight line cannot be drawn from O to l/k without passing through the singularity +1, the loop L x about l/k is formed by means of a curved line (Fig. 68) above the a?-axis and FIG. 68. a small circle about l/k. This loop can be deformed into the contour (Fig. 69) consisting of : (i) the ic-axis from O to 1 e y (ii) a small semi-circle c of centre 1 and radius e above the a>axis, described negatively ; (iii) the ic-axis from 1+e to 1/k e; (iv) a small circle C of centre l/k and radius e, described positively ; (v) the ic-axis from I/A; e to 1 + e; (vi) the semi-circle c described positively ; (vii) the ce-axis from 1 6 to O. FIG. 69. Since Lim (z l)f(z) = and Lim (z l/k)f(z) = 0, the integrals ->! s->lfk along (ii), (iv), and (vi) tend to zero with e. The integral along (i) gives K. As z passes round c, amp (z 1) 66] LEGENDRE'S FIRST NORMAL INTEGRAL 165 decreases by x, and (1 x) changes to (x l)e~ iir ; hence the integral along (iii) is CtX . -r-r f 1 where TT< K = Again, as z passes round C, amp (z 1 /k) increases by 2x, and (1 kx) becomes (1 kx)e 2iir ; hence (v) gives the integral dx jW* Finally, as z passes round c, amp (z 1) increases by x, and (x 1) becomes (1 x)e iir ; so that (vii) gives the integral dx =K _ Thus the value of the integral round the loop is,2K+2iK', and/(;s) returns to O with the value 1. It can be proved in a similar manner that the integral round the loop L 2 (Fig. 70) is 2K 2^K'. This follows more simply, Y, FIG. 70. however, from the fact that L 2 can be replaced by A, L 1? A' 1 , taken in succession : the value of the integral along this contour is then 2K-(2K + 2iK / ) + 2K = 2K-2iK / . Similarly, the contour C x (Fig. 70) can be replaced by A and Lj taken in succession ; so that the integral round C^ has the value 2K-(2K+2iU / )=-2^K / , and /(z) returns to O with the value +1. Finally, the integrals round the loop L 3 and the curve C 2 have the values -(2K + 2iK / ) and 2iK' respectively. Hence, if w denotes the integral || f(z)dz taken along a Jo straight line from O to z, with the initial value + 1 at O, the 166 FUNCTIONS OF A COMPLEX VARIABLE [CH. ix general value of F(z) is 2mK + 2niK'+( I) m w, where m and n are integers. The value of the integral when z is infinite can be found as follows. Let the integral be taken round the contour (Fig. 71) consisting of : (i) the straight line from O to z ; (ii) a semi-circle of centre O from z to z ; (iii) the line from z to O. Since this contour is equivalent to the contour C 2 (Fig. 70), the integral has the value 2iK'. But the integral along (ii) tends to zero when z tends to infinity (30, Th. I.), and [ f(z)dz= [* f(z)dz, J -z Jo since the final value of /(z) is equal to its initial value. Therefore, when z tends to infinity, I f(z)dz tends to the value -&K'; so that Jo f(z)dz = i If in the integral fl/A Jl * dx we put y=-^/(\kV)/k', where k' = *J(l k 2 ), we obtain K--P dy Jo It follows that K' is the same function of k' that K is of k. Inversion of the Elliptic Integral. In Example 2 of the previous section we deduced from the properties of the integral w = 1 -^ ^\ various properties of the inverse function z = sin w. o 66, 67] INVERSION OF THE ELLIPTIC INTEGRAL 167 Similarly, if iv=\ /f/1 2\?i 72 2\v z can ^ e regarded as a Jo v I \ ^ / \ ^ / / function of w, and from the properties of the integral those of the function can be deduced. We shall here make two assumptions : (i) that the function exists for all real or complex values of w ; and (ii) that the function is single- valued. These assumptions will be justified in Chapter XL The function is denoted by z = saw: from the general value of the integral it follows that Accordingly, sn w has two periods, 4K and 2^K', the one purely^ ^ real and the other purely imaginary, and sn(2K w) = sn^. -" Again, since f(z)dz= \ f(z)dz= w, it follows that Jo Jo z = sn ( w) = sn w ; so that sn w is odd. The properties of the integral also give : snO = 0, snK = l, Instead of sn w the notation sn (w, k) is frequently employed : k is called the Modulus and k' the Complementary Modulus of sn (w, k). Example. Shew that K' = log (4/&) + <(&), where <j>(k) tends to zero with L We have -f ( - J* WO-# 2 ) where y = kx. Hence = log2; from which the required theorem follows. 67. The Weierstrassian Elliptic Integral. Let t(;-l^ = f(z)dz, J ZQ ^liQref(z)={4>(z e l )(z e 2 )(z e s )}~^: here w = w corresponds to Z = Z Q) and one of the two values of f(z ) is selected as initial 168 FUNCTIONS OF A COMPLEX VARIABLE [CH. value. There are four branch-points of f(z) (Fig. 72), e l} e, 2 , e 3 , and oc . The loop L about oo , however, consisting of the line from z to f and a large circle described negatively, can be replaced by the loops L x , L 2 , L 3 , about e lt e 2 , e 3 , described negatively in suc- cession ; so that it is only necessary to consider the effects of these three loops. Let A 1= f(z)dz t A 2 = f(z)dz, A 3 = f(z)dz\ Jzo **9 ^ Z then integrals round the loops L u L 2 , L 3 , or Lf 1 , L.J 1 , L^ 1 , give the values 2A 15 2A 2 , 2A 3 , respectively. Two successive integra- tions round a loop give the value zero. Successive integrations round loops L,, and L 6 . give the value 2A r 2A S . Again, the description of an even number of loops brings f(z) back to z with its initial value f(z ), while an odd number brings it back with the value f(z Q ). Hence, if I denotes the integral I f(z)dz taken along a straight line from z to z, the general value of the integral is given by ^XXq \~ \ / ' where n lt n 2 , n^ are integers such that ?i x -f ii 2 + % has the value or 1 according as the number of loops described is even or odd. Now let n^ in 2 , n s = w\, so that either / n 2 = m 1 + m 2 or n% = ?7i 1 + m 2 -f 1 ; then either w-w = -2^- A 2 )-2m 1 (.A 3 - A 2 ) + I 67, 68] THE WEIERSTRASSIAN ELLIPTIC INTEGRAL 169 |% re, Again, if Wi=\ f(z)dz and oo. 2 = I f(z)dz, JeS J <>! A i A A r$\ A 2 A 3 = w 1 and A 2 A 1 = o> 2 ; hence either i# = u> +2??i 1 ft) 1 + 2??i 2 ft> 2 + I or w = Thus the inverse function z = (j>(w) is doubly-periodic, with periods 2co 1 and 2co 2 . Next, let the integral be taken along the contour consisting of the loops L, L 3 , L 2 , L 1? taken in succession. This curve encloses no singular point, so that the value of the integral is zero. But the integral round the large circle tends to zero as the radius tends to infinity ; hence = 2 /0)<i2;-2A 3 +2A 2 -2A 1 ; so that f f(z)dz = A 3 - A 2 + A r J*o Now take w = \ f(z)dz= A 3 -f A 2 A x ; then w= I f(z)dz. J t J CO Hence, if z = e lt w = 2??!^ + 2?^ 2 ft) 2 A 3 + A 2 Aj + Aj = < 2m jr w 1 + 2m. 7 w 2 + ^ or w = 2m t co x + 277i 2 w 2 A 3 + A 2 A 1 + 2 A 2 A x = 2?^ ^ + 2m 2 ft> 2 + 2o> 2 + w 1 . Therefore ^ 1 = 0(ft> 1 ). Similarly 6? 2 = 0(ft) 1 + ft) 2 ) and e 2 = <p(a). 2 ). Again, if W = ty H- 1, . te; = 2m x &>! -f 2m 2 w 2 + W or w = 2m Thus <f>(iv) is an even function of iv. It will be shewn in Chapter X. that <f>(w) is Weierstrass's Elliptic Function ^(w). It should be noticed that the signs of the two periods 2co l and 2o> 2 depend on the initial value selected for f(z ). 68. Elliptic Integrals in General. Any integral of the type JR(z, Jfydz, where R(#, y) is a rational function of x and y and Z is a polynomial of the third or fourth degree in z with real coefficients and no repeated factors, is called an Elliptic Integral. 170 FUNCTIONS OF A COMPLEX VARIABLE [CH. ix When Z is a cubic. the integral can be transformed into an integral in which Z is a quartic as follows. Let Z = (2 /3)(az* + bz + c), where /5, a, 6, c, are real; then, if *-=, which is an integral of the required form. Again, let R(#, y) = P(x, y)/Q,(x, y), where P(#, y) and Q(&, y) are polynomials in x and y ; then, since (\/Z) 2 ^, where p is a positive integer, is a polynomial in z, we can write where K(z), L(z), M(z), N(z), are polynomials in z. Now multiply numerator and denominator by M(z) where U(2) and V(2f) are rational in z. But U(2;) can be integrated by elementary methods. Hence we need only consider integrals of the type or where S (2) is rational in 2. Again, by the method of partial fractions, S(0) can be put in the form Hence the integral J{S(2;)/\/Z}cZ2; can be expressed linearly in terms of integrals of the types z n -, , dz dz and Now ( f^n := dz can be expressed in terms of the four integrals f* 3 7 f* 2 7 f* ,7 f^ 2 JTI^ J^* JTI^ J72- But 68] REDUCTION OF ELLIPTIC INTEGRALS 171 where Z = az' i -}-bz s +cz 2 +dz + e; therefore 1 T^- can be expressed in terms of the three integrals [z 2 dz [zdz [dz 1*7 ' \ /7 ' 1/7' J V " J \ J J V " Similarly, since dz (Z - OL) m ~ (z-OL) m + l v/Z j= can be expressed in terms of JO-afv/Z f dz Ccte C(z-oi)dz Kz-a.) z dz JO-(x)v/Z' JN/Z' J \'Z J VZ Thus every Elliptic Integral can be expressed in terms of integrals of the types dz dz Czdz Cz 2 dz 7z' J7z' J7I"' Again, since imaginary factors of Z always occur in pairs, Z can always be written a(z 2 +pz + q)(z 2 +rz + s), where p, q, r, s, are real. Now in the transformation z = (/-f<7 )/(! + )> ^ f and g be chosen so that the coefficient of in each quadratic is zero ; then Z will take the form It is always possible to find real values for in and n. For ^. n Q s * ps or f+q = 2+ - , fq = - -- ; r-p' J * r-p ' so that / and g are the two roots of the quadratic equation O - P)f 2 + 2 (s - #)/+ (ps - gr) = 0. Accordingly, if the roots are real, we must have (s-q) 2 -(r-p)(ps-qr)>0. (A) Now let the two equations x*+px + q = Q, x 2 +rx + s = 0, (B) have roots x : , x z , and x s , x, respectively; so that Then inequality (A) can be written (x l - a; 3 ) (x 1 - aj 4 ) (x. 2 - 172 FUNCTIONS OF A COMPLEX VARIABLE [OH. ix This inequality holds if one at least of equations (B) has imaginary roots ; for then the four factors consist of two pairs of conjugate complex quantities. Also, if both equations have real roots, the factors of Z can always be chosen so that Thus the inequality holds in this case also. It follows that real values of / and g, and therefore of m and n, can always be found. Accordingly, every Elliptic Integral can be expressed in terms of integrals of the types where Q = But and this integral can be evaluated by elementary methods. Also d . , f d? -ifl w->vo' and the last integral can be evaluated by elementary methods. Hence we need only consider the integrals There are four cases to be considered (we assume a In case (i) put =x/a, k = b/a', then the integrals are trans formed into integrals of the forms x*dx f dx fc^ dx In cases (ii), (iii), and (iv), make the substitutions l-a 2 f 2 = a 2 , l-W^ = x, and l + fe 2 f 2 = respectively ; then all these cases reduce to case (i). 68, 69] COMPLETE ELLIPTIC INTEGRALS 173 dx Now JVUi-^Xi-/^)} A^JVUI- | r //I _ 7.22 dx. " ""' ic Hence all Elliptic Integrals can be expressed in terms of Ellipti Integrals of the three types, dx The three definite interals, f* Jo(^ 2 - dx are called Legendre's Normal Integrals of the First, Second, and Third kinds. Example. Prove f 1 3^ + 2^ 2 , _ /r ? / Jo v^+^+i 1 * 3jo - But P ' dx Jo ^PT ' where ^ 2 /vs/s ^ 3 3 Jo 3 Hence the required equation follows. 69. Complete Elliptic Integrals. If in the First and Second of Legendre's Normal Integrals the substitution x = sin is made, they become 174 FUNCTIONS OF A COMPLEX VARIABLE [CH. ix respectively. In particular, if x = l, then <f> = ir/2, and these integrals become E = E(&, 7T/2) = a - A? sm*<p)dfa Jo which are known as Legendre's Complete Elliptic Integrals of the First and Second kinds. Similarly we write K' = F(F, 7T/2), E' = E(#, 7T/2). These functions can be expressed as hypergeometric series in k and k' : for, since k < 1, ^^ Similarly K' = F(J, }, 1, 7c' 2 ), E = -F(- J, J, 1, A; 2 ), 2j & The numerical values of K, E, K', and E' can be easily evaluated by means of these series, except when the value of k or k', as the case may be, is nearly unity, in which case the convergence is siow. Landens Transformation. If in the integral F(k, 0) we make the substitution tan(0 1 0) = &'tan0 or tan 1 = sin 20/(& 1 + cos , 7 1 k' k 7^7 T , . where A; 1 = -._, y / = /1 7 / V ;^<C^> we obtain l + /c ( 1 + /c )" and so that Y(k, 0) = Thus the integral is expressed in terms of an integral of smaller modulus. In particular, if = Tr/2, then fa = TT, so that 69, 70] LANDEN'S TRANSFORMATION 175 Accordingly, if the modulus of k is nearly unity, the value of F(k, 7T/2) can be deduced from that of F(/<; 1 , 7r/2) by means of this transformation. Bmmpl e. Prove and deduce from the example of 68 that P3.^+2^^. -1/1 TT Jo 70. Legendre's Relation. A relation can be established between the four quantities K, K', E, E', as follows. We have dK_W* ksm*<}>d<j> _1W 2 d<f> _K dk ~ J o ( 1 - /c 2 sin 2 0) 3 /' 2 ~ k] ( 1 - /c 2 sin 2 ^) 3 / 2 k ' 1z d sin0cos0 7c' 2 . 79 . But k 2 ~j- -rrtj r, . y = 7^ 79 . 2j vo/. 2 + v/( 1 & 2 sm 2 0). 2 2 23 / 2 Th, fore Hence ^^Jl _f C'/M &M tt & Accordingly, since 7c 2 + 7c' 2 = l, rfK E VK ,'ti , , , = T77T-/ + -TIT A JT^ ^ Therefore, interchanging 7^ and /c', we have Again, dK' = _E' + &K; //- 1 f-/ 2 7/1 1 f ff / 2 d0 E-K J(l k-sm 2 d))d<j> f . = - /.'J /^Jo l A; 2 sm 2 / ,_ K '>. CvK 1C f /\Y Accordingly, if W = KE r -h K'E - KIv, -^- = ; therefore \V is a constant. 176 FUNCTIONS OF A COMPLEX VARIABLE [OH. ix Now consider the value of (E K)K' when k tends to zero. Also Hence |(E- K)K' < (^& 2 +. so that Lim {(E-K)K'}=0. But, when fc = 0, K = 7r/2, and E' = l ; therefore W=KE' + K'E-KK' = |. COROLLARY. K and K' satisfy the equation where x = k 2 . This equation is known as the differential equation of the Quarter Periods of the Jacobian Elliptic Functions. EXAMPLES IX. rZ 7 1. If w= I 4> and if w is any value of w corresponding to z=z 0t shew that the general value of w for z=z is w +m>/2ir/4-f'N^7rt/4, where m and n are integers, such that m + n is even. ri +s 2 3<5?2, shew that, with the notation of the previous example, the general value of w is w -H773 + mr\/3/3, where m + n is even. 3. Find the most general value of / f , for any path of integration, JO ^(Z' + I) where the initial value of the integrand is unity. Am. wn + (-l) n log(l+\/2), .(w = 0,. 1, 2, ...). 5. Prove that, for the ellipse # 2 /a 2 +y 2 /6 2 =l, the length of an arc measured from the point (0, 6) in the clockwise direction is aE(e, <), where e is the eccentricity and ^'=asin <. 6. Prove that [Vws".r^=2v/2E(4=' \2 where cos ^ = cos 2 70] EXAMPLES IX 177 7. If a 1 > b- > c 2 , shew that y" 00 f^A 2 '- + A)}~v'O 2 - c2 ) 8. Shew that Ji where = sin 15. [Shew that the integral is equal to 2 r x o?y where A = (l - 9. Shew that where ^=cos!5. 10. Prove f flLy 2 / . TT\ - =Fsml5 ' 11. By means of the substitution ./; = (4 - T/ 3 )^ 2 , shew that dy . deduce that K' = V3K, where /fc=sin 15. ^ Stan 3 + 8 tan 2 # -2 tan 0+4 A H,, V(l + 2sin2^) - B prove that 13. Prove that the length of the lemniscate r = a\fcos20 is 2\/2F( - , J ) \V2 / 14. If -s' denotes the length of an arc of the hyperbola x* i \a, i y 1 \l> 1 =\ measured from the point where it crosses the ^-axis, shew that where J 15. Shew that, if K = ^ and K' = /'-, ''K liKK (*K ZK I 'rove tliat (K K'!\) satisfies the differential equation 4KK 1 -j^-_ t y. M.F. M 178 FUNCTIONS OF A COMPLEX VARIABLE [OH. ix 16. Shew that, if n>l, (i) n T k n Wdk=(n - 1) T "- 2 E'<tt ; Jo Jo (ii) (n + 2) r y^EW/j = (n + 1) T L M K'dL Jo Jo 17. If P is any point on that branch of. the hyperbola .r-/a 2 - which crosses the #-axis at A, shew that the difference between the arc AP and the portion of the asymptote cut off by a perpendicular on it from P tends to the limit as P tends to infinity. [Cf. Example 14.] 18. Shew that Y 2 dx _ _ * J* V{(i-^)(^' 2 +^ 2 )}" where ^= x /(l-^7 2 ). 19. Shew that dy where ky = *J\-x z . CH. X. 71] CHAPTER X. WEIERSTRASSIAN ELLIPTIC FUNCTIONS. 71. Doubly-Periodic Functions. A uniform function F(z) which has two primitive periods 2 and Q' is said to be Doubly- Periodic. For all values of z, so that where m and m can have any integral values. THEOREM. The two primitive periods Q and Q' cannot have the same amplitude. For, if they have the same amplitude, let Q = pe ie , Q' ==//**, and assume /o>/o'. Then, if tt" = tt-Q' = (p-p)e ie , IT is a period of modulus less than p. Let this process be repeated with the two periods Q' and Q" '; and so on. After a sufficient number of steps a period is obtained either of modulus zero or of modulus less than any assigned quantity. The first case cannot occur, however; for if u> denote the value of the two equal periods subtracted in the last step of the process, where p and q are integers ; but this is impossible, since Q and Q' are primitive periods. In the second case, if denote the period, the function {F(z) F(z )} has zeros at z and z +et>. Accordingly, F(0) has essential singularities at all points of the plane ( 22, Theorem I. Corollary 1). Such functions are excluded from consideration. Congruent Points. The points z + mil + m'l', where in and IK' may have any integral values, are said to be congruent to the point z. /'< ,'iod- Parallelograms. A parallelogram of vertices a, a + Q, a 4- 2', <t + fi + !Y, is called a period-parallelogram. It is sufficient 180 FUNCTIONS OF A COMPLEX VARIABLE [CH. x to study the behaviour of the function in one period-parallelo- gram in order to know its properties for the entire z-plane. If the whole plane be divided up by two sets of equi-distant parallel lines into a net-work of period-parallelograms, corre- sponding points of the parallelograms form a set of congruent points. An example of such a net- work was given in 37. 72. Elliptic Functions. A doubly-periodic function with no singularities in the period-parallelogram except isolated poles is called an Elliptic Function. It is convenient to choose the periods 2^ and 2o> 2 so that, as in 37, I^/o^) is- positive. Weierstrass's Elliptic Function. If we differentiate the series Z) = -s we obtain From this series the equations ?'(z + 2 Wl ) = p'CO, p\z + 2o> 2 ) = v'(z\ follow immediately; so that <@>(z) is an Elliptic Function. Again, integrating, we have Now let z = W then so that C = 0. Thus #> (z + 2^) = p(z). Similarly %>(%+ 2o> 2 ) = p(2;). Accordingly, p(z) is an Elliptic Function. COROLLAKY. If n is any integer, {$>(z)} n is an elliptic Junction. Note. The notation <p(z\ co 1 , w 2 ) is sometimes used instead of P(*> THEOREM I. The derivatives of an elliptic function are elliptic functions. For, if f(z + 2^) =f(z), f(z + 2w 2 ) =f(z), it follows that THEOREM II. An elliptic function must have at least one pole in a period-parallelogram. 71,72] WEIERSTRASS'S ELLIPTIC FUNCTION 181 For if not, the function would be finite at every point of the plane, and would therefore, by Liouville's Theorem, be a constant. Thus the function $(z) has poles of the second order at the origin and congruent points ; while at all other points it is holomorphic. The principal part at the origin is 1/z 2 . Similarly p'(z) has a pole of the third order at the origin, with principal part 2/z 3 . COROLLARY. If two elliptic functions have the same periods and the same poles, and if their principal parts at the poles are equal, they can only differ by a constant. Note. An elliptic function has an essential singularity at infinity : for it has an infinite number of poles in any neighbour- hood of infinity (cf . 48, Note). This holds true for all periodic functions ; e.g. cot z. THEOREM III. An elliptic function can have only a finite number of poles in a period-parallelogram ( 22, Theorem 2). THEOREM IV. The sum of the residues of an elliptic function f(z) at points in. a period-parallelogram is zero. Let y denote the parallelogram ABCD (Fig. 73) of vertices Fio. 73. ((, r/,H-2o) 1 , a -f 2^ + 2o> 2 , a+2a> 2 , drawn so that none of its sides passes through a singularity of f(z). Then the sum of the residues of f(z) in y is given by a+2o> 2 = 0. For example, the residues of p(z) and $(z) at 2 = are zero. COROLLARY. An elliptic function cannot have a single simple pole in a period-parallelogram. Oi'der of an Elliptic Function. The number of poles of an elliptic function in a period-parallelogram, a pole of order s / 182 FUNCTIONS OF A COMPLEX VARIABLE [OH. x being counted as s poles, is called the Order of the function. It follows from Theorem IV. Corollary, that the order of an elliptic function must be not less than 2. The two simplest types of elliptic functions are : (i) functions with a single pole of order 2, at which the principal part is of the form A/(z-oc) 2 , in each period-parallelo- gram ; q>(z) is a function of this type : (ii) functions with two simple poles of principal garts A/(z a) and A./(z (3) in each period-parallelogram ; it will be shewn in Chapter XL that the Jacobian functions snu, cnu, dn u, are of this type. THEOREM V. The number of zeros of an elliptic function f(z) in a period-parallelogram, where a zero of order r is counted as r zeros, is equal to the order N of f(z). For (31, Corollary 1) where y denotes a period-parallelogram. But, since f'(z)/f(z) is an elliptic function, this integral is zero (Theorem IV.). Hence Sr = 2s = N. Thus, since p'(z) has one pole of order 3 in the period- parallelogram, it must have three and only three zeros in the parallelogram. Now, substituting z = ^ in the equation $(z + 2ft*!) = $>'(z), we obtain ^'(^i) = %>'( o^). But from the series for $>'(z) it follows that %>'(z) is odd : hence ^'(^i) = 0- Similarly /(o> 2 ) = 0, ^(co l -\-(a z ) = 0. Thus the only non-congruent zeros of p'(z) are co lt w 2 , and o^ + o^. COROLLARY. Since the elliptic function {/(z) C} has the same - poles as f(z). the number of its zeros in a period-parallelogram will be N. Hence the number of points in a period-parallelogram at which /(z) = C is N. THEOREM VI. If the elliptic function f(z) has p zeros a 1? a 2 , ..., dp, of orders r l9 r 2 , ..., r pt and q poles 6 1? 6 2 , ..., b q , of orders 8 lt s 2 , ..., s q , in a period-parallelogram, m=l 7i where X and /x are integers. 72, 73] POLES AND ZEROS 183 /\ For, if y denote a period-parallelogram (31, Corollary 2), = - 2o> 2 Log 1 -f 2^ Log 1 Hence 2 r a l = l Example. Prove that ^= ^ irisa simple zero of This is an elliptic function in u of order 3, its only pole being at u=Q. Two zeros are u = v and tt, = w, so that the third must be congruent to -v w (Theorem VI.). Also (Theorem V.) each zero must be of the first order. 73. Relation between $(z) and p'(z). We shall now prove that $(z) satisfies the differential equation where # 2 and </ 3 are constants. Near z = we have ( (z- Accordingly / A - = But if ii is odd, 22 = ' therefore where fc- ' = 184 FUNCTIONS OF A COMPLEX VARIABLE [CH. x From this equation we derive the following equations : Hence, if <j>(z) denotes the function near 2 = 0, cj>(z) = Dz 2 + E,2 4 + .... Thus the elliptic function <f>(z) has no pole at the origin. But the origin is its only possible pole. It is therefore a constant (Theorem II. 72) ; and since 0(0) = 0, the constant is zero. Thus The quantities g 2 and g 3 are called the invariants of #?(z). It sometimes found useful to use the notation $(z ; g^g^) fo COROLLARY. By differentiating equation (A) we obtain : Thus every derivative of <p(z) can be expressed as a polynomial in z and 'z. Example. Prove that the function {jp(u)fl(u) + $P(u) 1} has five zeros, r=5 w i> U 2t U si u i u &> i n a period-parallelogram, sucli that S^-^SAwj + S/zwo, r=l where A and /x are integers. Verify that, if 2 = >(), these values of u give the five roots of the equation If /(z) = 0, equation (A) becomes Now we know (Theorem V. 72) that 0>'( w i) ^'(^2)' P / ( w i are all zero. Hence the three roots of this cubic in p(z) are e lt e 2 , 6 3 , where *e 1 = P(i) ^ = ^(^1 73, 74] THE ADDITION THEOREM 185 It follows that equation (A) can be written If the coefficients in equations (A) and (B) are equated, the following important relations are obtained : The Weierstrassian Elliptic Integral. Let z $>(w) : then, since Now when w = 0, z = cc ; therefore w= I J r The two branches of the integrand give equal and opposite values of w, which correspond to the same value of z, since is even. 74. The Addition Theorem. Consider the elliptic function The functions p(u + v), $>(u), and p'(u) have poles at u= v. u = 0, and w = respectively; while {p(u) <p(v)} has zeros at u=-v. Hence the only possible non-congruent infinities of f(u) are -it = 0, u= v. Near u = (), Accordingly, when u = 0, f(u) is finite and has the vahu- zero. 186 FUNCTIONS OF A COMPLEX VARIABLE [CH. x Again, let u = v + e ; then Hence /( 16) is finite when u = v. Finally, let u = v + e ; then Hence /(u) is finite at u= v. Thus f(u) is constant (Theorem II. 72). But when u = 0, f(u) has the value zero ; therefore This is the Addition Theorem for the Weierstrassian Elliptic Function. COROLLAKY. p(u-v)= -y Prove Duplication Formula. If u = -v + e, the addition theorem gives 74, 75] ADDITION OF A SEMI-PERIOD 187 Therefore, if e = 0, Example. Shew that The following three formulae can be deduced from the addition theorem : The proof is left as an exercise to the reader. Example. Prove 75. Properties of the Zeta Function. Integrating the equa- tion we have f (u + 2^) = f (w) + 2 where 2^ is a constant. Now, let u = - w l ; then f (o^) = so that ^i = Similarly f (t6 + 2o, 2 ) = % , where /7 2 = f (o) 2 ). It follows that f (u + 2???.^ + 2^w 2 ) = and that f ( mo^ + 7io) 9 ) = The Zeta function is not an elliptic function. It possesses, however, a sort of periodicity, and is called a Periodic Function of the Second Kind. In each period-parallelogram it has a simple pole congruent to 16 = 0. The residue at this pole is unity ; for, if we integrate we obtain 188 FUNCTIONS OF A COMPLEX VARIABLE [CH. x But, since g(u) is odd, C = ; therefore Example. Shew that f (2w) = 2f (u) + X ^' j& . 2 <? vv Again, let f (u) be integrated round the period-parallelogram y (Fig. 73) ; then r ra+2a>! |*a+2a> 2 I ^(^)^^ == l {(^) (^ ~l~ 2ft> )}cfat I {(^0 J v J a J a = 2-Tri, since there is only one pole in y. Thus This is Legendre's Relation for the Weierstrassian Elliptic Functions. THEOREM. Any elliptic function can be expressed linearly in terms of zeta functions and the derivatives of zeta functions. Let f(u) be an elliptic function of periods 2^ and 2o> 2 , and let a, 6, c, . . . , & be its poles in a period-parallelogram. Also let the principal parts of f(u) at these poles be _Ai _L ^2 , , A"i u - a "*" (u - a) 2 "*" ^ (w - .)"i ' B~D "D i JD O D. U /C (U , Then consider the function 75, 76] THE ZETA FUNCTION 189 This function is finite at all points of the period-parallelogram. Also = 0(w). (Theorem IV. 72.) Similarly <j)(u + 2a> 2 ) = 0(u). Accordingly, <j>(u) is a constant (Theorem II. 72); therefore (A) (74 1)! Example. Shew that 2f (2w) + 2r h + 2r? 2 = () -4- f (w + CD,) + {(u + w, + o> 2 ) + f (w + o> 2 ). 76. Properties of the Sigma Function. Integrating we have log{<r(u-h 2^)} =log{<r(t6)} + 2 % u + C; (cf.50) or o-(u + 2w 1 ) = CV(u)6 2 ''i it . * Now let u = w l ; then o-(fo) 1 ) = CV( a) l )e~ 27? i a) i, so that C'= -e 2l ii. Therefore o-( w + 2^) = Similarly <r(u + 2o> 2 ) = By the method of induction it can be deduced that ar(u + 2771^ + 27K 2 ) = ( l) wl '' l + The Sigma function is called a Periodic Function of the Third Kind. Near u = Q we have Ih-nce But Lim {-^ [ = 1, so that C = ; therefore ->0 v * J Thus 190 FUNCTIONS OF A COMPLEX VARIABLE [CH. x THEOREM. Any elliptic function can be expressed in terms of sigma functions. Let /(it) denote an elliptic function of periods 2u> lt 2a> 2 , having in a particular period-parallelogram zeros a lt a 2 , ..., a pt of orders m lf m 2 , ..., m p , and poles b lt b. 2 , ..., b q , of orders %, 7i 2 , ..., n q . Then consider the function We choose the a's and fe's so that 2?rfcct 2^6 = 0, replacing, if necessary, some of them by congruent points (Theorem VI. 72). Now (it) is finite at all points of the period-parallelogram. But (Theorem V. 72.) Similarly </>(u+ 2o> 2 ) = 0(u). Thus (Theorem II., 72), 0(u) is a constant ; so that */ \ / I ./ /i , For example, the function {^(^) ^(^)| has two simple zeros t>, and a pole of order 2 at u = 0; therefore In this equation let u be small ; then Hence, equating the coefficients of ,, we have TV so that P( tt )-P< ff )=- COROLLARY. If in equation (A) we put v = it + e> and make tend to zero, we obtain Example 1. Shew that n'f \ 76] THE SIGMA FUNCTION 191 Again, if equation (A) be differentiated logarithmically, In this equation interchange u and v ; then ' - {(u - v) - 2( v). Hence - = u + v - u - v . (B) 2 COROLLARY. If in formula (A) of -75 we make the substitution 2 #>(u) and similar substitutions for (u b), ..., f(t& &); then, since 2A 1 = 0, it follows that f(u) can be expressed as a rational function of >u and >'u. Example 2. From equation (B) deduce the addition theorem EXAMPLES X. 1. Find, the zeros of and shew that they are all simple zeros. Ans. v r w. 2. Find the poles and zeros of Ans. Simple poles, v, w\ simple zeros, 0, v-w. 3. If %>(z) is constructed with 2<it l9 2w 2 , as primitive periods, while $>i(z) is similarly constructed with 2d> 1 /72,.2(o 2} as primitive periods, prove that 4. Shew that V + ( (i) (ii) ^ 5. Shew that (i) 192 FUNCTIONS OF A COMPLEX VARIABLE [CH. 6. Prove 7. Shew that 8. Prove ff>(2y)-p(2w)=- 9. Shew that (ii) 10. . , ~ 11. Prove 12. Shew that i 13. If *()=f and ^(^ = shew that 14. Shew that 15. Prove 16. Prove (r(a + b)(r(a b)(r(a 17. Shew that ? (7 18. Shew that a-(2u) 19. Prove - d) - v(a + c)v(a - c)v(b + d)<r(b - d) u ^( u -<tfiMM- Q) 2 )<r(m-<u 1 + u> 2 ) o-(w 1 )cr((o 2 )fr((o 1 -f w 2 ) = 2- (0 - w)<r(w - u)<r tr 3 (?*) (T 3 (v) or 3 ( M; EXAMPLES X 193 20. Prove -z)<r(y-w) 21. Prove 22. Shew that 23. Shew that 24. If M + v + ?^ = 0, prove { f () + f (^ + f ( 25. Shew that 2^(2?*) = f (*) + f ( - 26. Shew that = P() + P u - o> 2 ). 1 p(w) tf*(w) 27. Prove f(w-*0-f(**-w) -(*-> / \ / V - 2v)ar(u - v)v(u - w) M.F. [CH. XI CHAPTER XL JACOBIAN ELLIPTIC FUNCTIONS. 77. The Values of <p(w) when o^ is Real and o> 2 is Purely Imaginary. Let (0^ = 0,^ ft) 2 = iQ 2 , where Q x and Q 2 are real and positive ; then i=0 m = -< The two terms in this bracket are conjugate complex numbers, so that g 2 is real. Similarly it can be shewn that #3 is real, and that, if w is real, $>(w) and <(p'(w) are real ; while if w is purely imaginary, $(w) is real and $?(w) is purely imaginary. Thus e 1 = p(Q x ) and e 3 = $>(il z ) are real ; also, since e% = e l e B , e 2 = ^(Q 1 + ^Q 2 ) is real. Hence the three roots of the equation 4# 3 g 2 x g s = are all real. Now consider the values of $(w) at points on the rectangle OABC (Fig. 74), where A, B, C are the points Qj, S^ + iQ^ ^Q 2 , respectively. v/k FIG. 74. (i) If w u is real, small, and positive, %>'(u) is, large and negative ; also, when u = l l , $>'(u) vanishes. Between these points on the real axis %>'(u) is continuous, and has no zero values ( 72, Th. V.). Accordingly, between and Q lt $>'(u) is negative ; so that $>(u) decreases continuously from +oc to e r 77] VARIATION OF p(w) 195 Now ^ /2 (^) Therefore since, as u increases from to Q x , ^(u) decreases continuously from +x to and $>(u) decreases continuously from +x> to e lt e l is the greatest root of ^x s g^cg s = 0. Again, p'(u) = <J{4<$ B (u) g$(u) g s } ; but between and Q 1? ^X^) is negative and 4{p(u) j}{|i(it) ^}{p(u) e 8 } is positive. Therefore Hence, if x <^(u), r dx U=\ , 3 J % v/ "j T?i/ " \j rt! provided a;^^. In particular, c?o? fee e, (ii) Let w = iv, where v is real ; then ' where 0^(^) = 40 3 (t;) As in (i) it can be shewn that e 1 = 0(Q 2 ) is the greatest root of 4x^g 2 x-\-g 3 = ) and that dx f* = J ^i Thus j_ or e 3 is the least root of 4# 3 # 2 # ^ 3 = 0, and Also, as v increases from to Q 2 , 0(v) decreases from -f x to e p so that ^(w ) increases continuously from oo to e 3 . Since ^ + 62+63 = 0, and e l '^>e f> '^>e 3 , it follows that e l must be positive and e s negative. (iii) Let w = u + il. 2 , where u is real ; the% since 196 FUNCTIONS OF A COMPLEX VARIABLE [CH. xi and p'(u + i& 2 ) are rea l- As u varies from to Q I} increases from e s to e 2> and p'(u+iQ 2 ) is positive. Therefore, between and f2 1? /7r so that Q 1= '" (iv) Let w = Qj + w, where v is real ; then, since iv) is real and ^'(Q^iv) is purely imaginary. As v varies from to Q 2 , ^(Q^w) decreases from 6 X to e 2 . Thus, if 0(v)= ^(Qj + iv), between and H 2 4>( v ) varies from e x to e 2 and 0'C^) ^ s positive. Therefore, since da; Hence Accordingly, as i; passes round the rectangle OABC, p(io) decreases continuously through all real values as follows : from + 00 at O to e 1 at A ; from e 1 at A to e 2 at B ; from e<> at B to e 3 at C ; and from e 3 at C to oo at 0. Let p be any real quantity, and let f be the point on the rectangle for which p(f)=JP- Then, since $>( )=p and <p(w) is of order 2, every point w such that #>(tt;)=^) must be con- gruent to f or Therefore, for every point within OABC, p(w) is imaginary or complex. Example. Shew that (i) (ii) 78. Geometric Application.* Consider the curve given by * Cf. Appell et Lacour, Fonctions Elliptiques, 68-63. 77, 78] GEOMETRIC APPLICATION 197 To each value of x correspond two non-congruent values w of the argument. But #>'( w)= ^'(w); hence to each point (x, y) on the curve there corresponds only one non- congruent value of w, and the curve is symmetrical about the aj-axis. Condition that three points should be 'collinear. Let M I} M 2 , M 3 , be the three points in which the line y mx c = Q cuts the curve. The corresponding values w l , w 2 , W B , of w are zeros of $>'(w) m$(w) c. Now the only pole of this function is at the origin, and is of order 3 ; thus Wi + ,,+,, where X and //, are integers. This relation is necessary, and it is also sufficient. For, if w l + w 2 +w z = 2\co l + 2yuft> 2 , let the line M^Mg cut the curve again in the point M' of argument w'\ then w l + w z + w' = 2X'o> 1 + 2yu'ft> 2 . Hence w 3 and w' are congruent, so that W coincides with M 3 . Tangents. If the tangent at 'w^ meets the curve again at ' w ' w+ 2w 1 = Thus w l = w/2 + Xo)! Accordingly, from any point ' w ' four tangents can be drawn to meet the curve in the four points whose arguments are Points of Inflection. At a point of inflection 3it; = 2Xft) 1 + 2^(o 2 ; so that ^ = 2Xa> 1 + 2 M a> 2 o Thus there are nine points of inflection with arguments 2^ 2a> 2 4fa? f 4o) 2 2 ~~ ~ ~3~' ~3~' 3 ' ~3~' 7 :> ' 3 ' ~~~3 ' ~ ~~3~ and they lie three by three on straight lines. Case in which ca l is real and w z is purely imaginary. Let then 2 where gj, e 2 , e 3 , are real, and ej 198 FUNCTIONS OF A COMPLEX VARIABLE [OH. xi As w varies along OA (Fig. 74) from to Q lt the point (x, y) passes up the right-hand branch of the curve of Fig. 75 from PIG. 75. y= o to A(e 1? 0). For values of w between Qj and y is imaginary. As w varies from Q 1 + iti 2 to iQ 2 , (x, y) passes from B(e 2 , 0) round BCD to D(c 3 , 0). For values of W between iQ 2 and 0, y is imaginary. The corresponding negative values of w give the other two arcs. There are only three real points of inflection, 0, 2Q x /3, and 2QJ3, the first being at infinity : they are collinear. Example 1. Shew that the necessary and sufficient condition that the six points whose arguments are w lt w 2t ...w 6 , should lie on a conic is Example 2. Shew that the necessary and sufficient condition that the 3n points w lt W<L, ... w sn , should lie on a curve of degree n is Consider the three 79. The Jacobian Elliptic Functions, functions: 78, 79] TRANSITION TO JACOBIAN FUNCTIONS 199 They satisfy the equations : ft) 2 ) = - <h( ft> 2 ) = 2ft),) = - 3 (u), ^ 8 (w + 2o> 2 ) = <fa(u). Again, by formula (A) of 76, Thus the two values of x/{^(X)~~ e i}> *J{p( u )~~ e z}t are the uniform functions <j>i(u), $> 2 (u), ty^u), respectively. If those values of the three functions are taken which are large and positive when u is small and positive, Now p / (^) = 2 Also, it is easy to shew that f'(u) = - 20 1 (w)0 2 (w)0 8 (w). (Cf. 76, Example 1.) Hence ^iX^) = ^( u }^( u )- Similarly <j> 2 '(u)= -^(u)^^), ^ 3 / (^)= - 0iO)02 ( u )- Next, let to! be purely real and co 2 purely imaginary, and denote them by Q x and iQ 2 respectively ; then, since p(u) ^ e l >- 6 2 > 6 3 , provided < it ^ Qj, - ViC^i) = 0, Similarly Accordingly, if these three functions will satisfy the equations : ^)= -S(tt), S 200 FUNCTIONS OF A COMPLEX VARIABLE [CH. xi S(0) = 0, C(0) = l, D(0)=l; ^ = 1, 0(00 = 0, Also S(, C(u), D(u), have simple poles at u = il 2 \ and is odd, while C(^), D(u), are even. Thus S(u), C(u), D(u), are elliptic functions of periods 2i0 2 ; 4Q lf 2Q 1 + 2iQ 2 ; 20! , 4i0 2 ; respectively. Now let S(u) = sn(v), where ^ and let so that & and /<;' are positive proper fractions such that k 2 + A/ 2 = 1 . Then sn(i>), cn(v), dn(?;), satisfy the equations : sn (v + 2K) = - sn v, sn (v -f '2iK') = sn (v) ; en (v + 2K) = - en v, en O + 2^K') = - en (v) ; dn (v + 2K) = dn v, dn (v + 2iK x ) = - dn (v) ; sn' (v) = en (v) dn (v) ; en' (v) = sn (v) dn (v) ; dn'(-u) = - 7c 2 sn (v) en <V) ; sn 2 (v) + en 2 (v ) = 1 ; k 2 sn 2 (v) + dn 2 (v) = 1 sn(0) = 0, cn(0) = l, dn(0) = sn(K) = l, cn(K) = (), / Also sn(v), cn(V), dn(v), have simple poles at v = iK'; and sn (v) is odd, while en (v) and dn (v) are even. Again, since <f> 3 (u) or <J(^(u) e s ) decreases continuously from to V( e i"~ 6 3) as u increases from to O 15 sn.(v) increases 79] THE JACOBIAN ELLIPTIC FUNCTIONS 201 continuously from to 1 as v increases from to K ; accordingly, if z = sn(v), where the positive value of the radical is taken between and 2=1. Hence dz and therefore sn (v) is the inverse function of 66. In particular, K - so that K is identical with the K defined there. Moreover, since *--mz*s=&=' r dx (77) Q 2 can be obtained from Q x by replacing e lt e 2 , e s , by e s , e 2 , I / Q g e lt respectively. Thus K' is the same function of /v/( J or k' that K is of A/( J ) or &; so that K' is identical with the K 7 of 66. Ve i-V These three functions sn(^), cn(f), dn(t;), are the Jacobian Elliptic Functions ; their periods are : 4K, 2iK'; 4K, 2K + 2iK'; 2K, 4iK'; respectively. Since ^(^) e 3 = ^(u), /? m ._, /? / \ ^1 ^Q (f\(f)j\ /> * ' e3 -sn 2 (.a)' //g g \ where v = ,J(e l e^).u and ^ = /y/( 2 3 j- This equation gives the relation between the Jacobian and the Weierstrassian elliptic functions. Example. Invert the function rdx v , \ H). 202 FUNCTIONS OF A COMPLEX VARIABLE [OH . xi Poles of sn(v), cn(v), dn(v). From the equation it follows that w^ = <f 2 = - i, ( 75) CQ)_ fa(u) D(u) so that Lim ;. r ^ 2 -^ = e ~ w^^ and Accordingly, if I is the residue of sn(t> ) at tQ 2 , the residues of cn(t;) and dn(v) at this point are II and ikl. The function su(v) has poles at iK' and 2K-f iK r , at which the residues are I and I respectively; it is therefore of order 2. Similarly cn(t>) and dn(v) are both of order 2. Note. The two periods K and iK' are not, like o^ and o> 2 in the case of the Weierstrassian functions, independent of each other : they are connected by the relations T 7-_ 1 dx ' ~ where Example. Prove (i) E== ( K dn 2 (w, lc)du ; (ii) E'= \*'&u?(u, k')du. Jo Jo 80. The Addition Theorems. Consider the functions of u, \(u + v) and cn(u)cn(u + v) cn(V): they both have periods 2K and 2iK', and simple poles at iK' and v + iK'. Hence they are of order 2, with simple zeros at u = and u = ' v ; so that where C is a constant. 79, 80] THE ADDITION THEOREMS 203 When u is. small, ...){cn( usn(v)dii(v) +...} cn(t>) 7 r ; - usu(v)-\- ... Now let 16 = 0; then C= dn(v); so that cn(V) cn(t6+ 1>) + sn (u) dn(v) sn(ic + v) cn(-y) = 0. If in this equation u and -y are interchanged, it becomes cn(t>) cn(u + v) -f sn(v) dn(u,) sn(i6 + 1?) cn(u) = 0. Hence, solving these two equations for sn(w + v), and writing c 1? cZj, s 2 , c 2 , cZ 2 , in place of sn(u), cn(u), dn(u), sn(^), cn(v), v), respectively, we have . 1 d l s l c. 2 c/ 2 Similarly In like manner, by considering the functions sn(u)sn(u-\-v) and dn(u)dn(y + i>) dn(v), it can be shewn that _ cZj d. 2 k 2 s 1 s 2 Cj <? 2 COROLLARY. If in these formulae v is written for v, they become Example. Prove 204 FUNCTIONS OF A COMPLEX VARIABLE [CH. xi Duplication Formulae. In the addition formulae make v = u; then, if sn(u), cn(u), dn(u), be written s, c, d, respectively, /tt . 2scd SEE sn(2tt) = = _, J. K S From these formulae the following can be derived : ^FTTC = F " D-C ' D + C 7c /2 1-D 7/2 l-C r+c = ^D^c Example. Shew that From the addition formulae it follows, since sn(K)=l, cn(K) = 0, that , , - dn(^) . dn(u) dn(u) Hence, if u tends to iK', su(K + iK') = jj cn(K + 'iK / )= -^, dn(K+iK / )-0. Now in the addition formulae put v = K + iK' ; then By repeated applications of these formulae the following can be derived : = cn(V), K x )= -dn(u); 80, 81] JACOBI'S IMAGINARY TRANSFORMATION 205 2iK / )= cn(u), Sample. Prove Again, in the addition formulae let i> tend to iK' ; then sn(u) Thus the residues of sn(u), cn(w,), dn(i(,), at iK' are I/A;, i/k, i, respectively. 81. Jacobi's Imaginary Transformation. Letx = sn(iu,k')', then Now put x = iy/J(I 2/ 2 ) ; then so that y = sn(u, /c). Thus sn(iu,k')=i su(H ' J ;\ cn(it, A;) To determine the sign let u tend to zero ; then, since T . sn(m, k') Lim . V ? 7 : = 1, n ^o t sn(u, /c) the + sign must be taken ; so that 1 . , k) Again, cn(m, k') = J{l sn' 2 (m, &')} = - To determine the sign let u ; thus cn(m, &') = -, j-. cn(u, k) Similarly du(iu, Jc) = / > ? . GU(U, k) Example. Shew that sn 2 (m, k') sn 2 ^, k) 206 FUNCTIONS OF A COMPLEX VARIABLE [OH. EXAMPLES XI. 1. Prove / . Jo 2. Shew that, if sn 11= sin <, rdu sn u dn u iTcn^ 3. Prove the following identities, in which D denotes 1 -J (i) sn (u + v) sn (u -v} = (c 2 2 - Cl 2 )/D = ( Sj 2 - s 2 2 )/D : (ii) {Icn(u + v)UIcn(u-v)} = ( Cl c 2 yiT>', (iii) {ldn(u + v)}{ldn(u-v)} = (d l d. 2 ) 2 ID; (i v) sn (u v) en (u qp v) = (s^d^ s 2 c 2 c 1 )/D ; (v) sn (u v) dn (u T v) = (s^c^ s^c^/D ; ( vi) 1 + en (u + v)cu(u-v)=(cS + c 2 2 )/D 5 ( vii) sn (u + v) en (u -v) + tm(u- v) en (u + v) = 4 Prove \ 2 / V 2 en ?6 + en v en 2 5. Shew that, with the notation of Example 3, D 3 -v}an(u-v) an"(u-v) 6. Verify the identity where S = sn (u + v) sn (u - v) sn (u + w) sn (u - w), C = en (u + v) en (u v) en {u + w) en (w w), D = dn (w + v) dn (u -v)du(u + w)dn(u- iv). 7. If S = sn u sn (u + K), verify, that : (ii) (iii) ( 1 _ v Deduce that 8. Shew that the function of M, snu cnu dn w(sn 2 y sn 2 2<;) + snv cnv dn v(sn 2 ^ sn 2 ^) + sn -^ en ?<; dn w(sn 2 ^ - sn 2 v), has periods 2K and 2i'K' ; and prove that u = iK f - v-w is a simple zero. 9. Prove that the function of u, sn 4 w(sn 2 v - sn 2 w) + sn 4 v(sn 2 w - sn 2 w) + sn 4 'w(sn 2 *i - sn 2 #), has periods 2K and 2^K' ; and shew that it has four simple non-congruent zeros, i\ w. xi] EXAMPLES XI 207 10. Verify that 11. Prove l-dn(2n) gV 12. Prove dn(w, k) = 'sn(K' - iK - iu, k). 13. Let w=sn 2 (z, k\ and let A, B, C, be the points K, K + ^K', ^K', respec- tively, in the 2-plane. Shew that, as z passes round the rectangle OOAB, w passes through all real values from - cc to 4- oc . If COAB is a square, what is the value of k ? Ans. 14. The coordinates of two points are connected by the equation ~ Shew that, as (#, y) describes the boundary of the rectangle COAB of the previous example, (X, Y) describes the complete boundary of a quadrant of a circle of unit radius. 15 Prove dn ( u + v) dn (u - v) k'~ cn(u + v) cn(u-v) afi 17 Shew that (i) Lim Bn " u ?=* ,..., T . udnu snu 1 2- nil) l/im 18. Establish the expansions en 2*= 1 - where \u\ < K'. 19. If the tangents from the point P on the cubic x=.yp(w\ y = p(w\ meet the curve in A, B, C, D, shew that the pairs of lines : AB, CD ; AC, BD ; AD, BC ; intersect at points Q, E, S, on the curve ; also shew that the tangents at P, Q, K, S, intersect at a point on the curve. [CH. xn CHAPTER XII. LINEAR DIFFERENTIAL EQUATIONS. 82. Continuation of a Function by Successive Elements. 03 Let P(z, a) denote a Taylor Series ^c n (z a) n with circle of o convergence C ; then, if z 1 is any point within C, this function CO can be expanded at z l in a Taylor Series ^jCn(z z^) n , which we o denote by P-^2, z^). The circle of convergence C x of this series will either touch C internally or lie partly outside C : in the latter case P^z, zj gives the continuation of P(z, a) in the part of Cj outside C (55, Th. II. Cor.). The two expressions P(z, a) and P^z, Zj) are called Elements of the function. The radius of C x will be the distance from z l to the nearest singularity of the function ; so that, if C x touches C internally, the point of contact must be a singular point. It may happen that no part of the circumference of C can be found, however small, which does not contain singularities of the function : in this case the function cannot be continued beyond C. If, on the other hand, the function can be continued beyond C, the process can be repeated with each new domain so attained. The aggregate of the elements thus obtained defines an Analytic Function. Note 1. If the only singularity of the function is at infinity, the original element gives the complete function. Note 2. If f(z) is holomorphic at infinity, the corresponding element is obtained by continuing /(1/f) to a domain of centre f-OL A particular point b can usually be approached by different continuations from a; and it is possible that the function may thus attain different values at 6. If the values are always the 82, 83] HOMOGENEOUS LINEAR EQUATIONS 209 same, the function is uniform. The case of multiform functions requires more particular investigation. Join a and b by a path L: on L take points a, z v z 2 , z 3 , ... , such that each point lies in the domain of the preceding one. Then the corresponding elements give a value of the function at each point on L. If no singularity lies on L, the points z i> z z> z s> " > can ^ e chosen so that, after a finite number of steps, a domain is reached which contains 6, and thus a value of the function at b is obtained. This value is independent of the set of points z lt z z , z s , ... , selected. For, let a set of points z ni , z n _ 2 , ... , z Ur , be interpolated on that arc of L which joins z n and z n+l ; then, if the elements corresponding to these points are employed in the process of continuation, the same value is attained at z n+l , since the arc lies entirely in the domain of z n ( 55, Th. II. Cor.). Now, any two sets of dividing points, z lt z 2 , z 3> ... , and z^, z 2 ', z s ', ..., can be combined, and other points, if necessary, interpolated between them, in order that each point of the new set may lie in the domain of the preceding one. Hence it follows that each of the original sets gives rise to the same functional value at b. Thus, if the function varies along a line which does not pass through a singularity, the set of values obtained at points on the line is always the same. Again, since the points z v z 2 , z 3 , ... , can be chosen so that each not only lies in the domain of the preceding point, but also in the domain of the succeeding point, it follows that, if the value at b be taken as initial value, and if the path L be retraced from b to a, the same set of values will be obtained at all points of the line. Finally, if any two paths L and L' are drawn from a to 6, such that no singularity lies between them, they will lead to the same value at b ; for otherwise the closed contour made up of L taken from b to a, and of L' taken from a to b, would enclose at least one branch-point of the function, which contradicts our hypothesis. 83. Homogeneous Linear Differential Equations. A linear differential equation d n w .d n ~ l w .d ll -*w 210 FUNCTIONS OF A COMPLEX VARIABLE [CH xn which involves no terms independent of w t is said to be Homo- geneous. We shall assume that the coefficients are uniform functions with no singularities except poles in the region con- sidered. A point which is an ordinary point for all the coefficients is called an ordinary point of the differential equation, while a point which is a singularity of any one of the coefficients is called a singularity of the equation. If f is an ordinary point, and if a is the singularity of the equation nearest to the interior of the circle z | = | a f | is called the domain of If the equation is of the first order, its solution is where C is an arbitrary constant. Accordingly, it is only necessary to consider equations of order higher than the first, We shall, indeed, confine our attention to equations of the second order; but the methods employed can be applied, with suitable modifications, to equations of higher order. THEOREM. In the domain of an ordinary point f the differ- ential equation c L^p(^ z \ ( - + q(z}w (A) possesses a unique integral w(z), which is a holomorphic function, and which, with its first derivative, acquires arbitrarily assigned values (the initial values) when z = f, Let Mj and M 2 be greater than or equal to the greatest values of |_p(z)| and q(z) on the circle z f| = R, where R<|a fl and a is the nearest singularity of the equation to f. Then (35, Cor. 1) the functions M *-jpb ^(z) = R satisfy the inequalities where n = 0, 1, 2, ____ The functions <p(z) and \fr(z) are called Dominant Functions, and the equation z) ^ is called the Dominant Equation. 83] EXISTENCE OF AN INTEGRAL 211 Now, if a function w(z) is holomorphic in the domain of f, it can be expressed in that region in the form of a convergent series c + c 1 (z-) + c t (z-&+..., (I) where < = !, *^>, (=0, 1, 2, ...). Tl : But if this function w(z) is an integral of equation (A), and if arbitrary values have been assigned to w(g) and w'( ), the corresponding value of w"(f ) can be obtained by substituting for 2 in the equation. Likewise, if the equation is differentiated repeatedly, and f substituted for z, equations are obtained for n = %, 4, 5, ... ; thus the coefficients c , c lt c 2 , ..., can be found. Similarly, if W(z) is a solution of equation (c), holomorphic within | z f | = R, where c n ' = , /&! and Now let I iv (g ) | and | to'(f ) I be assigned as initial values to and W(f) J tnen ' ^ rom equations (A), (B), (c), (D), and (E), it follows that, for all values of n, W (n) (f ) is real and positive, and Accordingly, if the series (n) can be proved to be convergent, the series (l) will also be convergent, and w(z) will be holo- morphic in the domain of f. Let 2-f=RZ; then . , (m) 212 FUNCTIONS OF A COMPLEX VARIABLE [OH. xn where c n " = R n c n ' ; and equation (c) becomes so that l- In this equation put Z = ; then, since we have Now let M x be chosen so great that RM X > 2 ; then so that c'nlc" n+l <^\.. H - T> TIT "D^lXf T) , C n 9 n-t-KtM-, . JtvlYl* But therefore Thus series (in) converges if | Z | < 1 ; hence series (n), and consequently series (i), converges if | z f | < R. Now, if z is any point in the domain of f, R can be chosen so that |z f|<R< a, |- Accordingly an integral w(z) exists, which is holomorphic in the domain of f, and is such that arbitrary values can be assigned to w(g) and w'(g). COROLLARY 1. The integral is unique. For, if any particular values are assigned to w(f) and w\g), only one set of values for <> C 3> C 4> ' > can ^ e deduced from equations (A) and (D). COROLLARY 2. The integral is of the form CQW I (Z)-\-C^JO^(Z\ where c , c x , are arbitrary constants, and w^z), w z (z), are integrals of the equation. For, by means of equations (A) and (D), all the constants c 2 , c 3 , c 4 , ... , can be expressed linearly in terms of c and c r Also, by making c and c^ zero in turn, we see that w^z) and w z (z) are integrals of the equation. Integrals at Infinity. To determine whether infinity is an ordinary point of the equation, the transformation z = l/ is employed. The equation then becomes a(MF\ w, 83, 84] SOLUTION BY INFINITE SERIES 213 so that it is necessary that p(z) + 2/z and q(z) should have zeros of orders 2 and 4 respectively at infinity. If this condition is fulfilled, holomorphic integrals w() or w(l/z) can be found. A nalytical Continuation of the Integral. Let f ' be any point in the domain of f, and let P(z, f ) be the element of the integral w(z) corresponding to the domain of f '. Then, since the function Jp( Z , n-j(*)^p(- n-?(*)p<*. r> vanishes at all points common to the domains of f and f, it vanishes at all points of the domain of f (55, Th. III.); thus P(X f) satisfies the differential equation, and has the initial values w(') and w'(') at f. Similarly it can be shewn that every element obtained from w(z) by analytical continuation satisfies the equation. 84. Solution by Infinite Series. An integral w(z) can be obtained by assigning values to c and c^ and then finding C 2> C 3> c t>'-> by means of equations (A) and (D) (83). In practice, however, it is usually simpler to proceed as follows : (i) if an integral in the domain of z = is required, substitute the series for w in equation (A), and equate the coefficients of powers of z ; a series of equations is thus obtained which enables us to deter- mine c 2 , c 3 , c 4 , ... , in terms of c and c l ; (ii) if an integral in the domain of any point a is required, apply the transformation z = a + to the equation, and use method (i) ; (iii) an integral in the domain of infinity can be obtained by applying the transformation z = l/f and using method (i); it is, 00 however, simpler to substitute the series ^c n /z n in the equation and equate coefficients. Note. The theorem proved in the previous section, and the method of solution just given, apply also to equations of higher order than the second. Legendre's Equation. Consider the equation -z 2 ) and q(z)= -7i(n+l)/(l-z 2 ); thus z = 214 FUNCTIONS OF A COMPLEX VARIABLE [CH. xn is an ordinary point of the equation, its domain being the interior of the circle z \ 1. Let <iv = c + c l z + c 2 z 2 +... be substituted in the equation ; then so that 2.1. 3.2.03-2^ + 71(71 + 1)^ = ^+2)^+1)^+2-^-1)^-2^ + 71(71 + 1)^ = 0, (,, = 2, 3, 4, ..,). Hence . 2 =- ~ 2.3 1J > Therefore w = c Q w l + c^v 2 , where w =F(-- n+l - z 2 ] w =* \ 2 *w / If 7i is an even positive integer, the first, and if n is an odd positive integer, the second of these series contains only a finite number of terms ; so that, if n is a positive integer, one integral is a polynomial. Now, if n is even, 2' 2 '2' (!) 2(2-l) n(n-l)(-2)(-8) ^ 1 " " '(z); (54, Cor.) while, if n is odd, m *i _l_ 9 Q \ \9 sF( - , + , -, ^ Thus, if n is a positive integer, one integral is the Legendre Polynomial ~P n (z\ which is also known as Legendre s Function of ike First Kind. 84, 851 LEGENDRE'S EQUATION 215 Example. Find integrals for Ans. Wl = l- + 6 - '2.5 7 2.5.8 I ,-fl __ 7! 10! 85. Fundamental System of Integrals. THEOREM I. The integral w(z) of equation (A), 83, cannot have a zero of the second order at any ordinary point of the 'equation, unless it vanishes identically. For if it has a zero of the second order at the point z, w(z) = and w'(z) = Q', hence the equation gives w"(z) = Q. Similarly, if the equation is differentiated repeatedly, it follows that so that the integral is identically zero. THEOREM II. If w (z), w 2 (z), w 3 (z), are integrals of the differ- ential equation holomorphic in the domain of f, a relation of the form c^z) + c 2 w z (z) + c 3 w s (z) = exists, where c lt c 2 , c 3 are constants not all zero. For if c lt c. 2 , c 3 , be chosen to satisfy the two equations w 3 (f ) = 0, (f) = 0, the integral w(z) = c l iu l (z) + c 2 w. 2 (z) + c^v 3 (z) and its first deri- vative vanish when z = . Hence, by Theorem I., w(z) is identically zero ; so that c l w 1 (z) + c z w 2 (z) + c s w 3 (z) = 0. DEFINITIONS. Two integrals are said to be linearly inde- pendent if their quotient is not a constant. Two linearly independent integrals are said to form a Fundamental System of Integrals. Such a system can always be obtained by making From Theorem II. it follows that if the integrals w (z) and w 2 (z) form a fundamental system, any integral can be expressed in the form c l w l (z)-i-c 2 w 2 (z) t where c and c 2 are constants. 216 FUNCTIONS OF A COMPLEX VARIABLE [CH. xn Again, if c l w 1 (z) + c 2 w 2 (z) = Q, then A(2) = 0, where A(z) = Conversely, if A(z) = 0, a relation C 1 w 1 for, if A(3) = 0, w i( z ) _ w z( z ) = exists: and the integral of this equation gives a relation of the type required. THEOREM III. If the integrals w^z) and w 2 (z) form a funda- mental system in the domain of f, A(z) cannot vanish in that domain. For let W 1 (^), W 2 (z), be another fundamental system ; then so that ' Jl ' (z) ' Wl(z) where '12 > The determinant D cannot vanish, since W^z) and W 2 (0) are linearly independent. But W-^z) and W 2 (z) can always be chosen so that, at any assigned point z in the region, W 1 () = l, W^) = 0, W 2 (2;) = 0, W 8 '(*) = l. Hence A(z) is non-zero at every point of the region. THEOREM IV. If two linearly independent functions w^z) and w 2 (z) are holomorphic in the neighbourhood of f, and are such that A(f ) =^= 0, a homogeneous linear differential equation of the second order can be constructed, of which they are integrals, and of which f is an ordinary point. For if the functions p(z) and q(z) are defined by the two equations w ^ -p(z) Wl '(z) - q(z)w 1 (z) = 0, then where p(z) = \(z)/A(z), q(z) = - A 2 (^)/A (z), ^'(z\ w 2 (z) , 2 Z 85] DETERMINANT OF A FUNDAMENTAL SYSTEM 217 Now the numerators and denominators of these two fractions are holomorphic, and A(f)^0; hence p(z) and q(z) are holo- morphic near z = g. Accordingly w^z) and iu 2 (z) are integrals of the equation w" =p(z)w' + q(z)w, of which f is an ordinary point. Example. Find an equation which is satisfied by w -I ^.Lif! 1 . 3 . 2 6 1 . 3 . 5 . 2 s Ans. w"= zw' + w. EXAMPLES XII. 1. Find integrals w lt w 2 , for w" + a 2 tv = Q, such that, when 2=0, ^ = 1, Wi = 0, w 2 = 0, w 2 ' = 1 . A ns. w l = cos az, w 2 = a~ l . sin az. Find integrals in the domain of 2 = for equations 2-10. 2 3 4 2 4.7 2 9 3. (2 2 - F) /' + zw' -w=0. Ans. w l = z, w 2 = V(^ 2 - 2 2 ). 4. (l+2 + 2 2 )w" + 2(l + 22)?^' + 2w = 0. Ans. ie 1 = ^ 5 , w a = -g. 5. (2-l)(2- 6. 7. (l- 8. (l-2 2 )w"-( W =F +M) _ 9. (1 - 2 s ) M' ^7M. ^=^,t^ = F(-, -f, , 2^), ^ 3 10. ^"-2 2 z^" + 22?^-2w=0. 22 s 22 11. Find integrals in the domain of 2=1 for 2(2-2)%'" A ns. w 1 =2(2 - 2), w 2 = 2(2 - 1) + 2(2 - 2) log {2/(2 - 2)}. 12. Find integrals in the domain of 2= - 1 for w" - (1 +z)w' -w=0. 218 FUNCTIONS OF A COMPLEX VARIABLE [OH. xii Find integrals in the domain of infinity for equations 13-15. 13. %" = (l-2s)3V + 2w. Ans. w^e 1 '*, w 2 =e~ 2/ *. 14. Z 4 w" + 2z 3 iv' + a 2 iv=0. Ans. w 1 = coB(a/z) J w 2 =sin(a/z). 15. z*(z*-l)w" + 2z(z* + I)w'-2w=0. Ans. iv^z^-l), iv 2 =z/(z 2 -!). 16. Find an equation which is satisfied by Ans. 17. Find an equation which is satisfied by iv 1 =z, w^e 1 . Ans. (z-l) w" - ziv + iv = 0. 18. Shew that, if n is a positive integer, the equation has integrals P(^) and P(^)log ( i!^+Q(z), where P(0) and Q(z) are polynomials of degrees n + 1 and n respectively. CH. xiii, 86] CHAPTER XIII. REGULAR INTEGRALS OF LINEAR DIFFERENTIAL EQUATIONS. 86. Integrals in the Neighbourhood of a Singularity. Consider a homogeneous linear differential equation of the second order, of which f is a singularity and of which w lt w z , form a fundamental system of integrals at z. Let z describe a closed circuit which encloses f but no other singularity of the equation, and let w l and w z be the analytical continua- tions of w l and w z obtained when the variable has completed the circuit. These two integrals ~w lt w z , form a fundamental system ; for, if not, a relation c 1 i(; 1 + c z w z = would exist. Consequently the function cjw l + c z w z would vanish at all points to which it can be continued (55, Th. III.); and therefore, retracing the circuit, we would obtain the relation c^w^c^w^O, which con- tradicts our hypothesis. Accordingly w^ = c ll w l + c lz w z , w 2 = c z] w l where D= _Now let W = Xw 1 + /*U' 2 , and choose the constants X, yu, so that W, the value attained by W after the description of the closed circuit, satisfies the equation W = pW, where p is a constant ; then Therefore, since w lt w z , form a fundamental system, c n -p, c 21 so that ''12 = 0. 220 FUNCTIONS OF A COMPLEX VARIABLE [OH. xm This is known as the Fundamental Equation belonging to the singularity If a root of this equation is substituted for p in equations (1), values of X and JUL are obtained such that W^pW. Neither of the p's can be zero, since D=/=0. If /o = l, the corresponding integral W will be uniform in the vicinity of THEOREM. The fundamental equation is independent of the original fundamental system selected. Let W x , W 2 , be any other fundamental system, and let so that the new fundamental equation is '21 6 12 , 0. &22-P N ow, if W 1 = ai l w l + a 12 w 2 , W 2 = a_ then W x = a n w t + a 12 w 2 , W 2 = a^^ + a 22 w 2 . = a n( c n w i + c iz Accordingly b n a u + 6 12 a 21 = anCj! + a 12 c 21 , Similarly Therefore (2) 21 a n + 6 22 a 21 = a a c u '12 a. 21 ^12' 1 > '21 Hence 600 p '2-2 C u /O, C, 21 C 22~~P Fundamental System associated with the Fundamental Equation. There are two cases to consider : (I.) when the roots of the fundamental equation are distinct, and (II.) when they are equal. 86] FUNDAMENTAL EQUATION 221 I. Let the roots p lt p. 2 , be distinct ; then there are two integrals W lt W 2 , such that Now let r Then, if O i = ( z -^r l) O z = (z-ft\ 1 = p^O-L , 9% = p 2 9. 2 ; so that W^ and W 2 /0 2 are uniform functions in the vicinity of f . Accordingly W, = (z - f)^W, W 2 = (z-fl^ 2 (3), where ^iC 21 ) an( i ^C 2 ) are uniform in the vicinity of The integrals Wj_ and W 2 are linearly ^ndependent. For, if not, an equation c x W 1 + c 2 W 2 = 0, and consequently an equation c i/iW 1 4-c 2 /o 2 W 2 = would exist. But these equations can only exist simultaneously if p 1 = p 2 , which contradicts our hypothesis. II. Let the roots be equal; then (c n c 22 ) 2 -f 4c 12 c 21 = 0. We distinguish between the cases: (i) when c 12 and c 21 are both zero ; and (ii) when they are not both zero. (i) In the first case p = c n = c 22 , and w l = pw 1 , w 2 = pw z . From equations (2) it follows that, no matter what system is originally selected, these equations hold. Accordingly ^l = (^-D r ^lW> ^ = (*-?) r ^2(3)> where V'l^)* ^2(2), are uniform in the vicinity of f, and (ii) In the second case, let W be the integral found to satisfy the condition W = pW, and let w be any linearly independent integral. Then w = c l W + c z w t and the fundamental equation becomes p _^ Q C l> C 2~ ( where <r is the quantity to be determined. Accordingly, since the roots are equal, c 2 = p ; therefore w = c^ W + piv. Now replace W by W lf where pW l = c l W, and write W 2 for Then W 1? W 2 , form a fundamental system such that 222 FUNCTIONS OF A COMPLEX VARIABLE [OH. xm Hence W^z where ^(z) is uniform near f. W Again, I but, if = - log (2; - f ), = + 1. Therefore sothat is uniform near f, Consequently where ^z * s uniform near 87. Regular Integrals. If the highest negative powers of (z ) in the Laurent Expansions for ifs-^z) and \^ 2 (2) are finite, the integrals W x and W 2 are called Regular Integrals. Now the quantities r = ^ . Log /o are not definite, but have values differing by integers. Hence, if the integrals are regular, the values of r : and r 2 can be chosen so that where a and b are non-zero. If a is the nearest singularity to these expansions are valid for |0 f |< | a f | . Thus w-'. | D'^i(^2^1og(2-D.J where r l r 2 is an integer or zero. For the first integral r x , and for the second integral the greater of the two quantities r x and r 2 , is called the Index at the point f It is only possible to carry out the theory completely when the integrals are regular ; and we shall therefore, in what follows, confine our attention to equations whqse integrals are regular. Condition that the Integrals at a Singularity should be Regular. If w is an integral of equation (A) of 83, a linearly independent integral can be found as follows. 86, 87] REGULAR INTEGRALS 223 Let w. 2 = wAvdz be substituted in the equation; then so that v = -'~ The integrals w l and w 2 are linearly independent. For, if not, therefore Hence, differentiating, we- have c. 2 v = 0. But v is not identically zero ; therefore c 2 = 0, and consequently c x = 0. Thus w^ w 2 , form a fundamental system. Also ' " jV, Since every other integral can be expressed linearly in terms of w l and w 2 , it is only necessary to find the condition that w l and We, should be regular. Now we can always choose w l so that v is free from logarithms. For, if w l is free from logarithms, while w 2 contains them, Thus and therefore v = ^= = ^ r - = v. dz\w l Hence v is uniform in the vicinity of Consequently z) or w^v is also free from logarithms. Again, if w l is replaced by W 1? where cw l = pW lt then W : = pWj , so that V is free from logarithms. O Now write w l and v for W l and V ; then MJ. = /o^j , w 2 = p (^ 4- ^ 2 )- Thus i^j and w z can be chosen so as to have the forms of formulae (A). 224 FUNCTIONS OF A COMPLEX VARIABLE [CH. xm In order to determine the index of A (2), three cases have to be considered. I. Let w t and w 2 be free from logarithms, and let r^r z . Then the index of A(z) or wfv is 2r 1 + (r 2 r 1 l) = r 1 + ?% !, since v = -r- (w z jw^. II. Let 10! and w 2 be free from logarithms, and let r 1 = r 2 . Then, by subtracting a multiple of w 1 from tu 2 , we can remove the first term of w z , and thus get Case I. III. Let w 2 involve a logarithm. If r z = r l} then where <f>(z) is holomorphic near f. Hence the index of A (z) is 2r l l=r l + r z I. If r 2 -<r 1 , the index of v is r 2 r x 1, so that the index of A (z) is r 1 + r 2 1. If r 2 > r x , then, adding w to w 2 , we get the case r 1 = r 2 . Hence in every case the index of A (z) is 7^ + ^ 1. Now p(z) = \(z)/A(z), q(z)= - A 2 (z)/A(z) (Theorem IV. 85). But a circuit about multiplies A(z), A^z), A 2 (0), by the same constant D (86); hence ^9(2;) and q(z) are uniform in the neighbourhood of f. Again, since A(0) has the index r + r z 1, A 1 (0) or j- A(0) must have an index =r l -}-r z 2 J and A 2 (is) or w 1 wfv %w l /2 vw l Witf an index > r x + T 2 3. Accordingly, in order that the integrals should be regular in the vicinity of the singular point it is necessary that the equation should be of the form * dw O where P(z) and Q(z) are holomorphic for |z f|<|a |. In the following section we shall prove that these conditions are sufficient. COROLLARY. If the integrals at infinity are regular, p(z) and q(z) must have zeros at infinity of the first and second orders respectively. The proof is left as an exercise to the reader. 87, 88] CONDITIONS FOR REGULAR INTEGRALS 225 88. The Method of Frobenius. If P(z) and QO) are holo- morphic for \z f [< a f|, a fundamental system of integrals can be found for the equation ?(z)div QQ) (1) such that both integrals are regular in the neighbourhood of If the origin is transferred to f, equation (1) becomes Z 2 w'' = 00(2X + \/r(z)w, (2) where 0(z) and ^(2) are holomorphic in the neighbourhood of the origin. Let w = z^c n z n ; then, if $(z)=^faP and \^(z) z$ (z) iv \fs(z)w n = where d = and d n = Hence, if all the quantities d , d^ d 2 , d B , ..., vanish, and o> if ^c n z n is convergent, w is a solution of (2). u The Indicial Equation. The equation in p, is called the Indicial Equation. From it can be obtained, in general, two values of p. If one of these values is substituted for p in the equations d l = 0, c 2 = 0, d 3 = 0, . . . , values for c i> C 2> c s> i are found in the form ,v V where H n (/o) is a polynomial in /o. If the roots of the indicial equation do not differ by an integer, none of the coefficients c lt c 2 , c 3 , ... , is infinite. If the roots are M.V. 226 FUNCTIONS OF A COMPLEX VARIABLE [CH. xm p l and p l + m, where m is a positive integer, then when p = p l , c m> c m+i> c m+2 are usually infinite. To avoid this we put C o = c (p-Pi) which makes c , c lt ..., c m _ 1 , all zero, and c m , c ro+1 , c m+2 , ... , finite, when p = p r Now assume cZ x = d 2 = c 3 = . . . = ; then z 2 w" - Z(f> (z)w' -\!,(z)w = z p c Q {p(p -l)-a p- 6 } , (4) Where C n = for 7i = l, 2,3,.... Let <p(z) and -fy(z) be holomorphic within and on the circle |z| = R. Then, if Mj and M 2 are the maximum values of <j>(z) and \/s(z) on this circle, Thus so that, if yn = +M | ^ j R" then Now Hence Vn _ \c n +M R Accordingly, if p is finite, and has not any of the values p l 1 , p 1 -2, ..., p 2 -l, p-2~ 2 > i wner e PI, p 2 > are the roots of the indicial equation, -> y CO CO Thus y]y n z n , and consequently 2 V 1 , converges if o o THE INDICIAL EQUATION 227 But if oc is the nearest singularity to the origin, R can always be chosen so as to include any point 0, such that | z |<| oc |, within the circle. Thus the series ^jc n z n is convergent for | co and ^v = z p ^ l CnZ n satisfies equation (4), if c lt c 2 , c 3 , ... , are given o by equations (5). Uniform Convergence of the Series with regard to p. Con- sider a region K in the p-plane bounded by the large circle \p\ = <r and small circles whose centres are those of the points p l I, p l 2, . . . , p. 2 1, p. 2 2, . . . , which are interior to this large circle. Then, if n = v>cr, for*all points of K, Now let v be taken so great that the last expression is always positive. Also let M denote the maximum value of M for the region K. Then, if -J R we have y n = C n , (n = v, j/ + l, + 2, ...). As in the case of the y's, we can obtain R' CO so that 2G H R' U is convergent if R'<R. Thus the series 71 V is uniformly convergent if | z \ = R' and if p lies in K. It is therefore holomorphic with regard to both z and p, provided that |zj<|tt.|, and that p has any finite values except p l 1, p 1 2, ..., /o 2 1, /o 2 ~~2, If, however, p z = p^-\-m and if CQ = C(P Pi), the point p 2 m is not excluded. The Fundamental System associated with the Roots of the Indicial Equation. There are three cases to consider. I. Let p l and p., differ by a quantity which is not an integer. 228 FUNCTIONS OF A COMPLEX VARIABLE [OH. XIH Then, if p is equated in turn to p l and p 2 , equation (4) becomes equation (2), and we obtain two independent solutions, II. Let the indicial equation have two equal roots p = p l ; then equation (4) becomes If this equation is differentiated with regard to p, it becomes -/>i) log s}. If in these two equations p l is substituted for p, it follows that w and both satisfy equation (2). Thus a fundamental system w l9 w 2 , is obtained for equation (2), where III. Let /o 2 = /i + m > where ?n is a positive integer. Then, if c is replaced by c(p p l ), equation (4) becomes Thus equation (2) is satisfied by the fundamental system Solutions free from Logarithms. If H m (/o) contains /Q ^ as a factor, c can be left unaltered, and both solutions will be free from logarithms. In that case w l will be of the form z pl P(z), where P(z) is a polynomial of degree =(m 1). 89. The Gaussian Differential Equation. The equation is known as Gauss s Equation, or the Hypergeometric Equation : it has singularities at 0, 1, oo . 88, 89] ASSOCIATED FUNDAMENTAL SYSTEM 229 CO In the vicinity of z = let w = ^c n g f+n ; then Thus the indicial equation is Xp-l)+yp Also, for 7i = 0, 1, 2, 3, ..., so that There are four cases to consider. I. Let 1 y be not an integer. Then, assigning to p the values and 1 y in turn, we obtain the fundamental system, w l = F(oi > 0, y, z\ w. 2 = z l -yF(oi- 7 +l, /3-y+l, 2-y, z). II. Let 1 y = 0. Then the indicial equation has two equal roots p = 0. Hence one solution is w l = c Q F(oi, /3, y, z). Again, = so that ' -l 2 3 r p+r+l' Thus the second solution is 2 = where *-' III. Let 1 y = ?7i, where m is a positive integer. One solution is ^ 1 = c F(oc, /3, y, z). Again, putting c i n ~ l X 1 r +-L.)- 230 FUNCTIONS OF A COMPLEX VARIABLE [CH. xm we have = ... "^ Hence the second solution is (a m)(oc m + 1) ... (a 1) (a m)(ot ?7i-fl) ... (a m + 9i 1) {m+n-l/ i S v^r~ 1 1 \ i-l 1 1 1 -- - r r If either oc or /3 is one of the numbers 1, 2, 3, ... , m, the terms involving log z disappear, and the second integral becomes ^-^(a-y + l, y + 1, 2-y, z), in which the hypergeometric factor is a polynomial. Since p + m is a factor of K m (p), this integral could also be obtained by o> putting p = 1 y in ^c n z p+n . o Let neither oc nor /3 have any of the values 1. 2, 3, ... , in; then, if w 2 is divided by the coefficient of log z, and a multiple of w l subtracted from it, the fundamental system can be taken to be w 1 = F(a. ) /3, y, z), ^ 2 = io 1 log^ + F 1 (a, (3, y, z\ where F/a, /3, y, z) 89] GAUSSIAN DIFFERENTIAL EQUATION 231 IV. Let 1 y be a positive integer. This case can be reduced to Case III. ; for the substitution w = z l ~ *W gives where a' = oc y-fl, ' = /# y+1, y' = 2 y; so that 1 y' = y 1 is a negative integer. Thus, if either oc or ft has any of the values 0, 1, 2, ... , y, the two integrals are F(oc, ft, y, z\ i-vF(oc-y + l, /3-y + l, 2-y, 2), where vanishing factors in the numerators and denominators of the coefficients of F(a, ft, y, z) are cancelled ; while if neither cc nor /3 has any of these values, the fundamental system can be taken to be , 2-y, z), 3-y + l, 2-y, z). Solutions Regular near z = l. The substitution z = 1 f gives Hence solutions regular near = 1 are obtained by replacing oc, /3, y, 2 by oc, $ oc + /3 + l y, 12, respectively in the integrals already obtained. For example, when y oc ft is not an integer, the solutions are F(oc, /3,oc+/3+l-y, 1-0), (l_3)y *F(y-/3, y-GC, y-a-/3+l, 1-0). Solutions Regular at Infinity. If we put 0=1 /f, w = a W, then Hence solutions regular at infinity are obtained from the solutions regular near = by replacing oc, ft, y, 0, by oc, 1-f oc y, 1+oc ft, 1/z, and multiplying by 0~ a . When a. ft is not an integer, the two solutions are iu l = z- a F(oi ) 1+oc-y, 1+OC-/3, 1/0), m 2 = 0-0F(& 1 + ^-y, l + /3-a, 1/0). The Differential Equation of the Quarter Periods of the Jacobian Elliptic Functions. If oc = ft = 1/2, y = 1, Gauss's Equation becomes iw = Q. (70, Cor.) 232 FUNCTIONS OF A COMPLEX VARIABLE [CH. xin It is left as an exercise to the reader to prove that solutions regular near 0, 1, oo , are : , , i , I o^/ter worked examples on differential equations, see Chapter XIV. 90, 9L EXAMPLES XIII. Find regular integrals in the domain of 2=0 for equations 1-16 : 1. 2zV+2i0'~(l+ a )w=0. Ans. *i = '+ + * + * + -' 2. z(\-z) oo fjn 4. zw" + w' w = Q. An s. u\ = 2 / , \.> > ... 5. (l+2)?//-2w = 0. ^?2S. 7^=2 + 2 , W 2 = ?<-'! log 6. ^' + 2-lw' + %' = 0. 7. 2^" - li' = 0. ^I?i5. Wj = f) ./ rr-: , o w!(^+l)l 2 2 /2 1\ z 3 /2 2 1 8, 89] EQUATION OF JACOBIAN QUARTER PERIODS 23S 9. 2 V + 0(1-0) w' - ( 1 + 20) w = 0. 10. ZW" + W' + mZW = 0. A ns. ll\ = 2 ( - 1 )" / t \2o2n ' 11. 0V + 4?' + 2zfr'=0. An*, i^ l/z, w z = l 12. 2 l" ' 13. 2 (l-)?^+2(l-0X-w=0. Ana. ^ 1 14, ^l -0 15. 92 2 w"-152w' + (3fo 4 + 7)w=0. ^7w. w x = 1/s cos 2 , w 2 = z lf * sin 2 1+<V3 l-<y. 17. Find regular integrals at infinity for 0W+(a z 18. Find regular integrals at infinity for 0%' ^l?s. %-j = -jj e 1/z2 , w 2 = 2?^ [CH. XIV CHAPTER XIV. LEGENDEE'S AND BESSEL'S EQUATIONS: EQUATIONS OF FUCHSIAN TYPE. 90. Legendre Functions. If the substitution z = l/g is made in Legendre s Equation ( 84), it becomes Let w=^c v ? +v ; then The indicial equation has roots p l = n, p 2 = n + l; and the second equation gives c l 0. Also (, = 0,1,2,...). In the first place assume that p 2 p l or 2n + 1 is not an integer ; then, if p = n, " --- 4 1 " while, if ! H L 2 n + I 3 2 ' 2 ' ^2' 2 90] LEGENDRE FUNCTIONS 235 Thus, if 2 n + 1 HO+l/2)z n + 1 Again, let 2n + 1 be an integer ; then n must either be an integer or half an odd integer. If n is an integer or zero, all the coefficients are finite. Hence both integrals are free from logarithms. In particular, if n is zero or a positive integer, 9/^!\2 w 'i = c 7^' P "< 2 )' (54, Cor.) V*Ji If 71 is half an odd positive integer, w 2 is the integral which does not involve log z, so that Q n (z) is an integral. If TI is half an odd negative integer, w l is the integral not involving log 2. But, in this case, since l/r(ii + 3/2) is zero when n + 3/2 is zero or a nega- tive integer, the first n terms of Q n (0) vanish, and therefore " _* 1_ ^ 2'2 n ' so that Qn(^) is again an integral. Accordingly, Q n (z) is an integral for all values of n. It is known asLeyendres Function of the Second Kind. P n (z) is the more important of the Legendre functions when |z|<l, and Q n (3) when!z!>l. Note. Thus far P n (z) has only been defined for positive integral or zero values of n, while Q M (z) has been defined for all values of n. Relation between Legendre 's Equation and Gauss's Equation. If in Legendrc's Equation we put z = 1 2f , we obtain which is Gauss's Equation with oi = n+l, /3= n, y = 1. Hence, in the vicinity of z = l, the two solutions are + 236 FUNCTIONS OF A COMPLEX VARIABLE [OH. xiv Definition of P n (z) for all Values of n. When n is a positive integer, Now it has just been shewn that this function satisfies Legendre's Equation for all values of n. Accordingly, for all values of n we define P n (z) by the equation COROLLARY. P n (z) = P _ n _ l (z). Example 1. If n is zero or a positive integer, shew that where the path of integration is taken so as not to pass through the point z, [Expand l/(z () in descending powers of z for \z\ > 1, and evaluate the coefficients by partial integration. The theorem holds if \z < 1, since the functions on both sides of the equation are holoniorphic.] Example 2. Use the series for Q n (z) to prove, for all values of n, the formulae : (i) (n Example 3. Use the expression P n (s)=F( -n, n+l t 1, ~^a~) to prove, for all values of ?i, the formulae : (i) (n + l)P n+l (z) - (2 + 1)P,,() + P M _ 1 () = 0, (ii) n-p n (z)=zP' n (z)-?' n -i(z}. Example 4. Shew that, for all values of , (i) [Use Ex. 2, (i), and Ex. 3, (i).] 91. Bessel Functions. The equation z*w" + zw' + (z* - n 2 )w = is known as BesseUs Equation, and its integrals are called Cylindrical Harmonics or Bessel Functions. 90, 91] BESSEL FUNCTIONS 237 The only singularities of Bessel's Equation are z = and z = oo . 00 To solve in the vicinity of = 0, put w = z?^c v z v ; then v = Hence c (yo 2 ^i 2 ) = ; ^{(p + 1) 2 ?i 2 } =0 ; c v {(p + vf n 2 } = c v - z , (i/=2, 3, 4, ...). The indicial equation is p 2 n 2 = : its roots are p l = n, p 2 = n. If pi p 2 is an integer, ??, must either be an integer or half an odd integer. The second equation gives c^ = ; so that c 3 = c 5 = c 7 = . . . = 0. Also r, f IV C <"2v \ L ) / where v = 1, 2, 3, .... There are four cases to consider. I . Let n be neither an integer nor half an odd integer. Then there are two independent solutions J n (z) and J_ n (z), where Z n / Z 2 Z 4 1 ~2 w n(w)l 2(2ri + 2) + 2^27?H-2)(27H-lt)~ "J J n (z) is holomorphic for all finite values of 2;, except possibly 2 = 0: it is known as Bessel's Function of the First Kind of order n. If n is a positive integer, J M (z) is an integral. J. n (z), however, is not a linearly independent integral. For, since l/IL(-n + v) = 0, where i/ = 0, 1, 2, ... , ^-1, II. Let 7i be half an odd integer ; then, since the coefficients in J n (z) and J- H (z) are all finite, these two functions are linearly independent integrals in this case also. III. Let n = Q, so that the roots of the indicial equation are equal ; then (-lys" 238 FUNCTIONS OF A COMPLEX VARIABLE [CH. xiv Hence co f i \ v 2v v ~[ ' Thus the two integrals are J oOO = . and v Y (2) is called BesseUs Function of the Second Kind of order zero. IV. Let n be a positive non-zero integer; then, if c Q = Hence = i^ 1v to + 2)... x f Vi 1 , A 1 . y> . 1 :+v_ _i \~ l p~n+2r ~{p + n + 'lr ^p + n + ~lr ^p + Sn Accordingly, if p = n, z n z- 1 z n 2 2. 4. ..'2n.'2A. ..(2^-2) 91] RECURRENCE FORMULAE 239 If these two integrals are multiplied by 2 n ~ l (n l)!/c, they become J n (z), and (-1)- /W+frf * 1 Subtracting ;;( y + ^ + H j) Jn(z) from the latter integral, we obtain the integral, Y M (2) = J(z) log z - ^ V (ll ~~ v ~ where ^r)i|++...+ v (r-l l 2, 3, ...), and Y n (z) is called Bessel's Function of the Second Kind of order n. Recurrence Formulae. We leave as an exercise to the reader the verification of the following formulae : (i) 2J n '(s)=j n . 1 (2)-J n+1 (2); (ii) J '()=-J 1 (); J w () as a Function of n. Let 2 1 = R, | w | = N ; then, if m is an integer such that m N> 1, and if T ,v 00 then | T v (z) \ = M p . But 2 ^ ^ s convergent ; consequently, by m+l o Weierstrass's M Test, 2 ^(2) is uniformly convergent if 240 FUNCTIONS OF A COMPLEX VARIABLE [OH. xiv \ Now R and N can be chosen so large that these regions enclose any assigned finite points z and n. Accordingly, for all finite values of z, except possibly 2 = 0, J n (z) is a holomorphic function of n. The Bessel Function G. n (z).* It is sometimes found convenient, instead of J_ w (z) or Y n (z), to take as the second solution of Bessel's Equation the function where the limiting value of the expression on the right-hand side is taken for G n (z) when n is an integer. Now Also PW+8l TYn vs , v _ v^4i/!r(- so that / v ~~ - 1 ; I o 7T . VI -n+2v ^(-n + v). If 7i is a positive integer, let p = n; then Accordingly, if n is a positive integer, O f T / \ : -r , 2 COS 717T *Cf. J. Dougall, Proc. ^IM. Math. Soc., Vol. XVIII. p. 36. 91] THE FUNCTION G n () 241 The verification of the following formulae is left as an exercise to the reader : (i) G_ tt () = e**G M (z); (ii) 2G n '() = G n _ 1 (*)-G n+1 (); 27? (iii) ^G.(z)=G,,. 1 (*)+G l , +1 (X THEOREM. If P(z) and Q(z) are any solutions of Bessel's Equation, they satisfy a relation of the form where C is a constant. For, if the substitution iv=z~ l ' 2 W is made in Bessel's Equation, it becomes Consequently {x Hence, integrating, we have a * - F() Q(*)= j- For example, 1 sin mr . T(n) = - and therefore J n ( 2 ) j'_ n (z) - J' n (z) J_(,-)= - 2 The reader can easily deduce that : ~J(*)J-ti-lW sillWTT. .=2 (iii) G n The Zeros of J n (s). If n is real and greater than - 1, all the zeros of J n (z) are real and distinct, except possibly z=0 ; this can be shewn as follows. We have - Thus, multiplying the first equation by J n (fiz), and the second by J,,(ou), and subtracting, we have 242 FUNCTIONS OF A COMPLEX VARIABLE [OH. xiv Hence, if K(?0> -1, (<X 2 - Therefore, if O=OL and 6 ft are distinct zeros of J M (0c), Again, let /3=cx + ; then If this equation is divided by e, and is then made to tend to zero, the equation becomes Hence, if 0=oc is any zero of J M (#c), except ^=0, THEOREM I. If n is real and greater than -1, J n (z) cannot have any purely imaginary zeros. For and the latter expression cannot vanish if y is real. THEOREM II. If n is real and greater than -1, J n (z) cannot have a complex zero. For if z=p + ig is a zero, where p and q are real, z=p iq must also be a zero ; hence P- I x J n { (p + iq)x } J n { (p iq)x}dx=Q. Jo But if n and x are real, the integrand is positive ; and therefore the integral cannot be zero. Thus the theorem must hold. Accordingly, if n is real and greater than 1, it follows that every zero of J n (z) must be real. THEOREM III. If n is real and greater than - 1, 3 n (z) has no repeated zeros except possibly 2=0. For if 2=0. is a zero, so that J'(o-) + 0- Thus Jn() has no repeated zeros. 91,92] THE ZEROS OF J H (z) 243 THEOREM IV. If n is real and greater than 1, J n () and J n +i(z) have no common zeros except possibly 2 = 0. This follows from the formula 92. Equations of Fuchsian Type. Equations whose coefficients are meromorphic in the entire plane, and which have their integrals regular in the vicinity of all their singularities, are called Equations of Fuchsian Type. If the singularities are a lt a z , a 3 , ..., a n , and infinity, the equation is of the form dw P 2n . 2 (z) = dz* (z - Oj)(z - 2 ) ... (z - c^) dz^(z- a l ) 2 (z - a 2 ) 2 . . . (z - a n where P n _ 1 () and P 2 n-2( 2: ) are polynomials of degrees n 1 and Zn 2 respectively ( 87). If infinity is not a singularity, the equation is of the form dz* (z-a l )(z-a 2 )...(z-a n ) dz(z-a 1 )\z-a 2 )\..(z-a n ) z where the coefficient of the highest term in P n -i(z) is 2 (83, p. 212). THEOREM. The sum of the indices associated with the singularities a lt a 2 , ... , a n , oo , of the equation of Fuchsian Type is Ti- 1. Let P_z and let ^ (z) (z a t ) (z a z ) . . . (z a n ). Then the indicial equation for the singularity a r is p(p 1) = p ""^ ^-\- terms independent of p. \js \a r ) Accordingly, if the roots of the indicial equation are p l and Now, by the theory of partial fractions, 244 FUNCTIONS OF A COMPLEX VARIABLE [CH xiv Hence, integrating round a large circle which encloses a lt a 2 , ... , a n , we have since Lim 2 s^so Thus the sum of the indices at a lt a z , ... , a,,,, is n + A. Again, put z=l/f ; then the equation becomes Thus the indicial equation is so that p l + /o 2 = 1 A. Hence the sum of the indices is n 1. COROLLARY. If infinity is not a singularity, A =2, and therefore the sum of the indices is n 2. 93. Riemann's P-function. We shall now investigate the conditions that the equation should be completely determined if the n + 1 singularities a lt a 2 , ... , a n , oo , and the corresponding indices, are assigned. There are 3n l constants to be determined in the equation. The assigning of the singularities a l} a 2 , ... a w , oo , simply deter- mines \fr(z) and the degrees of P w _!(2) and P 2n _ 2 (z). The assign^ ing of the 2rz, + 2 indices determines only 2n+I constants, since the indices must satisfy the condition that their sum is nl. Thus n 2 constants remain to be determined ; so that, if 7i = 2, the equation is completely determined. Similarly, when infinity is not a singularity of the equation, there are n 3 constants to be determined ; so that the equation is completely determined if n = 3. Consequently, in both cases, if there are three singularities, and if the indices are given, the equation is completely determined. By means of the transformation zhcb a 92, 93] RIEMANN'S P-FUNCTION 245 the equation with singularities h, k, oo , can be transformed into an equation with singularities a, b, c. The equation can therefore always be put in the form d 2 w ( f g h }dw where f+g + h = 2. Let the indices at a, b, c, be X and X', /UL and //, v and j/, respectively, where X + X' + yu + yu' 4-^ + ^=1. Then, since the indicial equation at a is 1 +/= x + X', l=- \\'(a - b)(a - c) ; so that /=X + X'-1. Similarly g = /m + // 1 , m = /*//(& ~ c )(^ ~ a ) h = v+i> 1, w= i/i/(c a)(c 6). Hence the equation can be written /uL /u.'l vv'\ dw ~~ ~ dz*^\ z- z-b z-c J dz + \~ z-a ~ +J z-b ~ + z^c~ w (z-a)(z-b)(z-c) Now for simplicity assume that X X', /U. JUL', v v, are not integers ; then, if P A , Py, P M , P M ', P,,, P v >, are integrals corresponding to the indices X, X', /UL, /UL', v, v', any branch of any integral of the equation can be expressed in any of the forms Riemann denotes such a function by {a, b, c, \ X, yu, v, z V; X', yu', I/', J and it is called Riemann s P-function. If either X and X', yu and //, or i/ and i/', are interchanged, the differential equation remains unaltered. Likewise the three columns can be inter- changed without altering the equation. Again, if the function is multiplied by (z a)*(z c)*(z 6)-" p , the indices at a, b, and 246 FUNCTIONS OF A COMPLEX VARIABLE [CH. xiv c, become X + or and X' + a; /JL <r p and // o- /o, v + p and i/ + p, while the branches of the function remain holomorphic at all other points, including infinity. Also the sum of the indices is still unity. Consequently a, 6, c, } I a, b, c, \ x, M , i/, z V=P^ x+cr, M-O p, j/+/o, z y. \ / / / i ^ / - / / , X , p , v , J ^X+o-, /m <r p, v +p, Again, the transformation f_za c b ^~ z b c a changes a, b, c, into 0, oc , 1. When the latter three points are the singularities of the equation, the function is denoted by p/X, M, , z \ IX, JUL , V, where K is a constant ; thus Z '(i-*y?{ x ' * ": .j-pf^*' "C'l* "-?' 4 tX, /x, i/, J LA+O-, /x o- p, ^+/>, The differential equation determined by P( J M ; ^ 2 1 is ob- IX, M, j/, J tained by putting a = 0, c = l, and making b tend to infinity; it can, by means of the equation X-f-X'-f A* +/*'+" -fV^lj be put in the form XX' - (XV ^ In particular, the function P( 1 ' ^' 3 2) satisfies ! . \1 y, P, y . p, / the hypergeometric equation Note. Since ' K -y, ft y a ft where a = X + /* + ^ /3 = X + /x r + 1/, y = l-X x + X, it follows that the P-function can always be expressed in terms of the integrals of the hypergeometric equation. 93] INTEGRALS OF HYPERGEOMETRIC EQUATION 247 The Twenty-four Integrals of the Hypergeometric Equation. The solutions corresponding to the indices 0, 1 y, at z = are (89), F(OC, ft, y, 2), Z l -y~F(oL y + 1, /3-y+l, 2-y, 2): we denote them by W 1 (0) and W 2 (0) respectively. Alternative forms for W 1 (0) and W 2 (0) are obtained as follows. We have / 0, oc, 0, 0, y 8, GC + /3 y, \ _ y> y _ a> 0, '/ Thus (l_z)Y--0F(y-a, y-/3, y, 2) = C t W^ + C 2 W 2 (0) . But, since the function on the left-hand side is uniform at 2 = 0, C 2 = ; hence In this equation let 2 = ; then 1 =C r Therefore W^ = (l-z)y-*-PF(y-OL, y-/3, y, 2). It follows that In like manner alternative forms can be found for the regular integrals at infinity and 2=1. Again, the six transformations, when applied any number of times in any order, change the points 0, oo , 1 , into the same three points in different orders. By means of these transformations new forms can therefore be obtained for the integrals. For example, jg 0, 0, oc, 0> a> 0> v , y-ft/S- = C(l-z)-F(a, y-/3, y, ^J. 248 FUNCTIONS OF A COMPLEX VARIABLE [CH. xiv In this equation let z = ; then C = 1. Hence These two expressions for W/ ) are valid if R(z)<l/2. We have thus obtained four different forms for W/ ). Similarly four different forms can be found for W 2 <>, W^ 1 ), W 2 < ] >, W^ 00 ), W 2 <*>. These twenty-four forms for the integrals of the' hyper- geometric equation are : III. IV. V. VI. VII. VIII. IX. X. XL XII. XIII. XIV. XV. XVI. XVII 2-y, z) a-y + 1, z-l z z-l z a-y + 1, a- 93, 94] ANALYTICAL CONTINUATION 249 xvm. =^-*(- XIX. = z-l' 1-z XX. = z l ~ XXI. W 2 (> = 2-fl XXII. =2 a - Y (z-l) Y - a - p F( 1 -a, y-a, - XXIII. = ( l)-*Fnft y-a, /3-a+l, y-^ Y Y T^r 1 ~ V / l\V~^~lTTI/3 _1_ 11 rv /Q rv _L 1 \ * 21 Relations of the form where r=l, 2; s = 0, 1, oo; ^ = 0, 1, oo ; hold between the six functions W/ ), W 2 ( ), W^), WgW W^"), W 2 ^\ One of these relations is given in Example 4 of 63, and the others can be found by similar methods. Example. Shew that, if y -.-/? is not an integer, the analytical con- tinuation of F(OL, /?, y, z) in the vicinity of z = \ is [Apply Ex. 4, 63, to form III. of W^.] 94. Spherical Harmonics. The equation is called Legendre's Associated Equation. The integrals of this equation are called Spherical Harmonics of degree n and rani- m. The most important cases are when n and in are positive integers, such that in =11. If 771 = 0, the Harmonic is a Legendre Function or Zonal Harmonic. If ?7i = l, 2, 3, ... , 7i 1, the Harmonic is a Tesseral Harmonic. If ?7i = 7i, the Harmonic is a Sectorial Harmonic. 250 FUNCTIONS OF A COMPLEX VARIABLE [CH. xiv Let the substitution w = (z 2 1)* MI W be made in Legendre's Associated Equation ; then Again, differentiating Legendre's Equation m times, where m is a positive integer, we obtain Accordingly, if m is a positive integer, two independent solu- tions of Legendre's Associated Equation are -r/ m P (?\ m rlmC\ (y\ p m(~\ _ / 2 2 _ 1 \ 2 a r n\Z) r\ w / \ _ / 2 i \2 a ^>n\Z) dz m ' ^ n ( ' dz m ' These functions P n m (z) and Q n m (2) are known as Legendre's Associated Functions of the First and Second Kinds respectively. To make them uniform a cross-cut is taken along the real axis in from oo to +1, and that branch of (z 2 1) is chosen which is real and positive when z is real and greater than 1. If m and n are positive integers, and m = n, - 120 > ( 93, Form II. of W (7^ + m)! /2-i\i / 1_ 2 \ = ,/ \I'("TT) F Ti+1, w, ?n + l, ^r 77i! 71 771! \2; 4-1 / \ 2 / If m>7i, P n m (z) = 0. Similarly, if in is a positive integer, then for all values of n, _(-iy*( z 2 +" M + m+2 ii + m + l 31 941 LEGENDKE'S ASSOCIATED FUNCTIONS 251 Again, let the equation obtained from Legendre's Associated Equation by means of the substitution w = (z* l)- * W be differentiated m times ; then Hence, if 971 and n are positive integers, and if m = ti, two independent solutions of Legendre's Associated Equation are and 771 f 3 f 2 f z = ( 2 ; 2 -l)-2 ... J OO J <X> J 00 Since the four functions P n m (z), P~ m (z\ Q n M (z), Q; m (2;), satisfy the same equation, they cannot all be independent. The relations connecting them are found as follows : "1 m flnm f=-W- s r a ('-) 'Sa^-l)- " l ; * 3 1 (93, Form II. of W 252 FUNCTIONS OF A COMPLEX VARIABLE [CH. EXAMPLES XIV. 1. Shew that, for all values of n : (i) (ii) (iii) (iv) 2. Shew that (i) Qo(*) [Use Ex. 1, 90.] 3. If n is zero or a positive integer, shew that positive circuits about z=l and z= -1 decrease and increase Q,(2) respectively by Tn'P^). [Use Ex. 1, 90.] 4. Use the formula of Example 1, 90, to prove the formulae of Example 2, 90, for positive integral values of n. 5. For all values of n, shew that (* ~ l){Qn(*) Pn(t) ~ P n (z) <&(*) } = C, where C is a constant. [Substitute P n (z) and Q,,(z) for w in Legendre's equation, multiply the two equations so obtained by Q n (z) and P n (z) respec- tively, subtract, and integrate.] 6. If n is a positive integer, shew that 7. If n is a positive integer, prove : (i) n{P n (z)Q n ^(z)-Q n (z)? (ii) 8. Shew that (i) z3n (ii) zG n 9. Prove that (i) (ii) 10. Shew that: (i) J^z)^ sinz ; (ii) J_^(z) Deduce that, when n is half an odd integer, J n (z) can be expressed in terms of elementary functions. 11. Shew that : (i) ~{z n 3 n (z)} = z n 3n-i(z) ; (ii) {z^ n (z)}^ -z~ n J n+l (z) xiv] EXAMPLES XIV 253 12. Shew that ,]m (i) 2' ?L J H (Z) = C J-(*) ~ C, J- m + 2 + -.+(- 1) M C W J, l + m (4 (ii) 2- ^ G ()= CoG^s) - cA^+oCO + ...+(- l)"^G, i+m (4 where c , c 1} ... , c m , are the coefficients in the expansion o 13. Establish the expansions : (i) ?J n - 1 (z) = n,J n (z)- 14. Shew that J n (z) is the coefficient of f" in the expansion of e ^ f* in powers of ^. 15. Establish the expansions : (i) co8(am^)=J (ii) sm(2sin^) = 2 [In Ex. 14 put f=e ifl in turn.] 16. If n is a positive integer, prove J n (z) = - rco$( IT JO [Multiply expansions (i) and (ii) of Ex. 15 by cosw^ and sinnO, and add.] 17. Shew that ' - __ [Multiply together the expansions of e \ *' and e V f /, and find the term independent of .] 18. If R(w)> - J, shew that [Expand cos(2cos<^>) in powers of 0, and evaluate the coefficients.] 19. Solve 2?0"+w=0. Ans. 20. Solve 2%"-2 21. Solve 22. If i, w, A % , are positive integers, and k<m, k<n, shew that (i) wl 254 FUNCTIONS OF A COMPLEX VARIABLE [OH. xiv 23. If n is an integer, shew that Equate the coefficients of f in 24. If n is an integer, shew that 25. Deduce Gauss's Theorem ( 61) from the Example of 93. 26. Shew that, if y - a. - /?<0, F(a, ft y, *)_r(. Sf <i-,r-'~ r while if y a. )8=0, [For the second equation apply Ex. 4, 63, to Form III. of W/' ( 93).] 27. Shew that, in the domain of the origin, every solution of Legendre : s Associated Equation can be put in the form CH. xv. 95] CHAPTER XV. SOLUTION OF DIFFERENTIAL EQUATIONS BY DEFINITE INTEGRALS. 95. First Method of Solution.* If Q(z) and L(z) are quadratic and linear functions of z respectively, and K is a constant, the equation Q can be put in the form w = 0, (A) where R(z) is linear in z. We shall confine ourselves to the case in which the factors of Q(z) are distinct. If the function I 0(f)(f z) A+1 fZf is substituted for w in Jc equation (A), then f te=; so that I #(^){X(f ) x " 1 Q(f)+(f 2^E(f)}df=sO, (B) Jc Accordingly, if 0(f) satisfies the equation equation (B) becomes Jordan, Cours d' Analyse, t. in, p. 240. 256 FUNCTIONS OF A COMPLEX VARIABLE [CH. xv Xow, equation (c) gives where p and q are constants, and f a, f &, are factors of Thus so that where D is a constant. Accordingly, is an integral, provided that either (f )*( &) 9 (f zY vanishes at both extremities of C, or else C is a closed curve such that this function (or the integrand) has equal values at the initial and final points. Let P be any point of the f -plane, and let A, B, and Z, denote loops drawn positively from P about a, 6, and z. Also let A, B, Z, denote the values of the integral taken round these loops, with M as the initial value, in each case, of the integrand at P. Any of the contours ABA" 1 B~ 1 , AZA- 1 Z~ 1 , BZB~ 1 Z~ 1 , where, for instance, the first denotes the loops A, B, A" 1 , B - 1 , described in succession, can be taken as path of integration C. For, if ABA~ 1 B~ 1 be taken, the final value of the integrand is equal to its initial value multiplied by ^irip glniq ^ - Snip g - 2niq == J . and similarly with the others. Let the values of the integral taken round these three contours be denoted by [AB], [AZ], [BZ], respectively. The value of [AB] can be found as follows. The loop A gives the integral A, and brings the integrand back to P with the value Me 2>rip . Thus the loop B gives the integral e* wip B, and the final value of the integrand is e^^+^M.. After describing the loop A~ l , the final value of the integrand is e 27 " 5 M, so that the corresponding integral is e 2niq A ; similarly the integral due to the loop B' 1 is B. Thus [AB] = (1 - e 2 ^) A - (1 - 95] BRANCH POINTS OF THE INTEGRAL 257 Similarly [AZ] = (1 - e 2 -*) A - (1 - and [BZ] = (1 - e 2 ^)B - (1 - Hence (1 -e 2 ^)[AB] + (l -e 2 ^)[BZ] + (l - e 2 ^)[ZA] = ; so that a linear relation exists between the three integrals, as is to be expected. Any two of these integrals, say [AZ] and [BZ], can be taken as the fundamental system. The Branch Points of the Integral. When z is fixed, the path of integration can be deformed without altering the value of the integral, provided that it is not made to pass over any of the points a, 6, z. If z varies continuously, the integrals will also vary continuously, provided that the path of integration is Fio. 76. deformed, when necessary, so as to avoid passing through the points a, 6, z. If z describes a contour about a, the loops A and Z (Fig. 76) must be deformed into loops A' and Z'.* Now Z' is equivalent to ZAZA^Z' 1 and A' to ZAZ" 1 or ZAZ~ 1 A~ 1 A. Thus, if Z' and A' are the values of the integrals taken along Z' and A', A'=-[AZ] + A. * This can be effected as follows : (i) deform Z into Z lf so that z passes round a to z l ; (ii) deform A into A' ; (iii) deform Zj into Z', so that z moves from % into its original position. M . 1 '. R 258 FUNCTIONS OF A COMPLEX VARIABLE [CH. xv Accordingly, [AZ] is transformed into [AZ]', where [AZ]' = (1 - e***)A? - (1 - e^p)Z' (D) Similarly [BZ] becomes [BZ]', where [BZ]' = [BZ] + (e 2 ^ - l)e 2 *[AZ]. Thus a is a branch point of both integrals. Similarly it can be shewn that b is a branch point. Infinity is also, in general, a branch point ; but a circuit about it can always be replaced by circuits about a and b. 96. Gauss's Equation. If in equation (A), 95, Q(z) = 2-z 2 , R(3) = (a-y + l)-(oc.-0 + l)2, X=-a-l, then a = 0, 6 = 1, > = . y + 1, # = y /3; thus the equation becomes Gauss's Equation, and has the integral where C is so chosen that the initial and final values of the integrand are identical. A second integral can be obtained by interchanging a and /3, and a third by putting 1/f for The latter integral is Employing the notation of 62, we can write one such integral, f where the initial point lies on the real axis between and 1, and the initial values of ^~ 1 and (l-f)v-^- 1 are real and positive. If z describes a closed contour enclosing z = but not z = 1 , the singular point 1/2 will describe a closed contour enclosing z = and z = l; and therefore the contour of the integral need not altered. Accordingly, for values of z which lie in a simply- connected region enclosing 2 = 0, but not enclosing 2=1, the integral is a uniform function of 2. 96, 97] THE HYPERGEOMETRIC FUNCTION 259 Now let |z|< 1, and choose that value of (1 z )~* which has the value + 1 when z = ; then ,0+, 1-,0-J ^ )} -)2 w , (62) , y-/3)F(oc, y, 2). Note. The expression given by this equation for the function F(oc, /3, y, 2) as a contour integral is valid for all values of z. Example. Prove F(o., /?, y, 2) = (1 - z) ~ a F (OL, y - /?, y, " [Put = l-C.] Again, consider the integral where the initial point is on the straight line joining f = to f =z, and the amplitudes of f/2 and (l f/2) are taken to be zero at this point; while that branch of (1 f)?"^ 1 is taken which has the value 1 when f =0. From formula (D) of 95 it follows that when 2 describes a closed contour about 2 = 0, the integral is multiplied by e~ Zvi y. Now let =zZ; then the integral becomes fO+,.1 + ,0-,1-) Z a ~v(l Z)~ a (l zZ)?-' , a-y + 1, 2-y, 2). This equation gives an expression for the function zi-yF(oc-y+l, /3-y + l, 2~y, 2), which is valid for all values of 2. 97. Legendre's Associated Equation.* If in equation (A), 95, Q(z)=l-z 2 , R(z)=- 2(w + 1)2, \=- w -m-2, then a = 1, 6 = 1, p = ^ = w- + 1 ; *Cf. Hobson, Phil. Trans., Vol. 187. 260 FUNCTIONS OF A COMPLEX VARIABLE [OH. xv thus the equation becomes (l-z 2 )i</'-2( and has the integral f (f*- JC where C is a suitable contour of integration. Hence ( 94) is an integral of Legendre's Associated Equation. The Function P n m (z). Consider the function f(z+, =(z 2 -i)H where a cross-cut is taken along the real axis in the z-plaiie from 1 to x to make the function uniform in z, and the amplitudes of zl and z+1 lie between TT and +TT. Let A (Fig. 77), a point in the f -plane on the straight line joining f =1 to f=z, be taken as initial point; and let the initial amplitudes of f 1 and f+1 be and 0', where these are the angles (between TT) which the lines joining f=l and f = 1 to A make with the positive -axis. Also let the initial value of amp(f z) be -.(TP ,^ so that amp(f z) is zero for points on the contour at which f z is a positive real quantity. Thus if z lies on the cc-axis to the right of +1, the initial values of amp(-l-l), amp( 1), and amp (f z) are 0, 0, and TT, respectively. 97] THE FUNCTION P n m (z) 261 Now let f 1 =(z 1)Z ; then the initial value of amp Z is zero. Again f+1 = 2U + ^-ZJ. But when f=l, amp(f+l) = 0; hence amp (l -4-?-^ Zj is zero when Z = 0. Also where amp(l Z) is initially zero. Thus ('?-!_ 1 \i-l IJL ) 2 3 I/ fl+, 0+,1-,0-) / 'yl \w Z(1-Z) ^l + ^Zj /^__1\T( 1 +. o+, i-,o-) \ y In particular, if 771 = 0, rjH-, !+,-,!-) (f 2 l)(f 3)-*- 1 df=2 w+2 7re' wl sin?i'7r 1-^ N so that ( 90) Now, if ?7i is a positive integer, then ( 94) 2 __ iiH r( Z +,i+,2-,i-) J. (f 2 - 262 .FUNCTIONS OF A COMPLEX VARIABLE [OH. xv But this function satisfies the differential equation for all values of ra. Hence, for all values of n and m, P n m (z) can be defined by any one of the equations i _ (93) COROLLARY. P? n _ ,(z) = P(z). Example 1. Shew that (Z+, 1+, Z-, 1-) f(Z+, 1 + ) / f(z+, 1+) deduce that Function Q, n m (z)> Again, consider the function ~ l where a cross-cut is taken along the real .axis in the z-plane from 1 to oo to make the function uniform in z. Let the origin in the f -plane be taken as initial point ; and let +1 and 1 have initial amplitudes 2?r and TT respectively, so that they will both have amplitude zero when f is real and greater than 1. Also let the initial value of amp (f 2) be ampz TT. Then, if | z l Sin 717T / ,2r+3\ 2 / (Exs. VIII. 7) ) t , , ( 62,Ex. 2). 97] THE FUNCTION Q H W (z) 263 Now, if m is a positive integer ( 94), O ^-(- But we have just shewn that this function satisfies the equation for all values of n and in. Hence, for all values of n and m,'Q w m (z) can be defined by either of the equations _ < - vli ,/w+m+2 XN -I- 1 < 2 n+] f(-l+, X ) COROLLARY. By applying the formula ( 93) F(a, ft, y, ) = (!-)' F( y -a, y-ft y, f). we obtain the relation A Second Expansion JOT Q n m (2). Consider the function There are two cases to consider, according as I (z) is positive or negative. Let A (Fig. 78), the initial point, be on the straight line joining f=l to =z, and let this line make an angle with the positive -axis. Also let the initial values of amp(f+l) and amp(f z) be and (TT 0) respectively. Then if f+l=(z+l)Z, the initial value of ampZ is zero. Also f z = (z+l)(Z 1), so that the initial value of amp(Z l)is TT. Again, since f 1= 2(1 -- ~^\ an d since, when f = 1, \ 2t J 1) has the value TT in the first case, and the value TT 264 FUNCTIONS OF A COMPLEX VARIABLE [OH. xv in the second case, -1 has the value 2e iir l--- -Z) when I(z) Lt is positive, and the value 2e~ iir (l -- 9^ z ) when I(z) is negative. -i FIG. Hence the given function has the value e niri ~~ m \z+l/ fi+,o+,l-,o-) Z n (I-Z)- n - m - dZ m) x F^-TI, Ti + 1, 1 -m, ^t 2 ), ( 96), according, as I(z) is positive or negative. Now let L, M, N, be the values of f(f 2 - : l) n (f-2)" w " m ; 1 d^ taken round loops from f=0 about 1, 1, z, respectively; the initial value of amp( 1) will be TT or TT according as I(z) is positive or negative. Then ( 95) : C(z+ , 1+, p.,, f(-l+,+ 97] A SECOND EXPANSION FOR Q n ' n (z) 265 Denote the integral in the last equation by W l ; the initial value of amp(f 2 1)" in this integral is mri, according as I (2) is positive or negative. Again, let W 2 denote the integral 1 > ({ 1 -i) r tf-*)**' r *' l 4t in which the initial values of fc.rnp(f+l) and amp(f 1) are 2?r and TT respectively; then an:p (f 2 1 ) n is mr initially. Hence e llni W 2 = e^ niri W l ; so that W 2 = e-' l7r ^ n7ri (L-M), according as I(z) is positive or negative. But Hence, since 1 ~ fz+, 2*+! it follows that O m/ 2 \ = _ x e ^m according as 1(2.) is positive or negative. COROLLARY. From the equation it follows that (c . p . 263) -(J.tl)'- F (-,,.,, +I , 1+ ,,!J-"). Example 2. Shew that 266 FUNCTIONS OF A COMPLEX VARIABLE [OH. xv 98. Second Method of Solution. Differential equations of the type (az+a')w"+(bz + b')w' + (cz + c')w = Q (A) can be integrated as follows : Substitute w= 0(0 6 ^f i n equation (A) : then Jci ( <l>(S)e*{(a^+b+c)z+(a'?+b'S+c')}dS=(). (B) JC Hence, if <p(g ) satisfies the equation (a'? + b'+c')<t,(f) = ^{(a?+bS+c)<t,(t)}, (c) /* .^7 equation (B) becomes jf.O()d=Q, Jc <* ~ where 0(f)#(f)(a*+ &+ e), Also equation (c) gives Thus <J>()e^ z dg is a solution of equation (A), provided C is so Jc chosen that 6(z) regains its initial value at the final point. 99. Bessel's Equation. In Bessel's Equation ( 91) put w = z n W; then This is an equation of the type considered in the previous section. Accordingly, since, in this case, =f J W c is an integral, provided (9(z) or e*(f 2 + l) w +* regains its initial value at the final point. Hence, if f is replaced by ig, a solution of Bessel's Equation is 'c where C is a suitable contour. 98, 99] EXPRESSION FOR 3 n (z) 267 Expression for J n (z). Consider the integral [-i+, +i-) where the initial point lies on the -axis between 1 and +1. Let the initial amplitudes of f+1 and f 1 be 2-Tr and TT respectively, so that each of them has zero amplitude at the point where f crosses the f-axis to the right of = 1. Then f(-l+,+l-) /A z \v fl *(?- !)"*#= V J J v=0 = - g* cos ^r(^ (Exs. VIII. 7) (62, Ex. 2) Hence COKOLLARY. If Example. Prove Expression for G n (z). There are two cases to be considered. CASE I. Let 7r/2 = = 7r/2 , where <j> = amp z. Then consider the integral taken along the contour C of Fig. 79 from infinity back to infinity. Both extremities of C approach infinity in a direction making an angle ?r/2 with the positive -axis ; so that iz is real and negative, and therefore 6(g) tends to zero at both 268 FUNCTIONS OF A COMPLEX VARIABLE [OH. xv extremities. The amplitudes of f+1 and f 1 are chosen so that they vanish at the point L, where the curve crosses the positive ^-axis. If necessary, deform the path so that, at every point on it, ; then -1 O +1/L FIG. 79. Now put X = e^zg, so that the initial and final values of amp X are and 2?r respectively ; then if W = : where the integral is taken along the contour of Fig. 60 (61). Hence ( 62, Ex. 2) Thus J_.(,)- 99] EXPRESSION FOR J_(z) 269 Next, let R(> + i)>0, and - Tr/2 < ^ 7r/2 ; then/deforming the path C into a contour (Fig. 80) consisting of a line through FIG. 80. f=l, which makes an angle ?r/2 with the positive -axis, described from x to 1, the -axis from 1 to 1, and this path reversed, we have S^ !>-*" where in the latter integral f 1 and f+1 have the amplitudes corresponding to the first description of the line from oc to 1. Hence so that G M (z) = e 8niri cos nTr Now let f l = e ^ 27r4 ' X, so that X is real and positive; then since, when f=l, amp(f+l)= 2?r, 270 FUNCTIONS OF A COMPLEX VARIABLE [OH. xv Thus - e- CASE II. Let 7r/2 ^ ^ 37T/2. Consider the integral f J where C is the contour of Fig. 81, and the amplitudes of f and f -|- 1 are chosen to be zero at L. If | f | > 1, yy iy ( } FIG. 81. so that, applying the transformation X = we obtain in the same way as before, to this integral, Next, let RO + J) > 0, and 7r/2 ^ < 3-7T/2 ; then, deforming C into a contour (Fig. 82) consisting of a line through f =1, which makes an angle Tr/2 (/> with the positive -axis, described from oo to 1, the -axis from 1 to 1, and this path reversed, we have nfl 99,100] EXPRESSION FOR G n (z) 271 " where in the latter integral 1 and f-fl have the amplitudes corresponding to the first description of the line. Hence so that Fio. 82. Now let f l = e X, so that X is real and positive; then Thus Accordingly, if R(TI + J)>0, this formula holds for all values of z such that 7r/2<<< 37T/2. 100. Asymptotic Expansions of the Bessel Functions. In the formula obtained at the end of the previous section, let z\e~ i * = g, so that ( is real and positive ; then V-l niri .'{ 7T\ / (^r^^'H^-' provided K(7i + |)>0, - 7r/2 < < 37r/2. Hence 272 FUNCTIONS OF A COMPLEX VARIABLE [OH. xv where the integral is taken along a line making an angle y with the positive real axis such that ?r/2 < r\ < ?r/2. Thus Now let g=ue ir> , so that u is real and positive; also let u) = \fs(u) + ix(u), where \Js(u) and \(u) are real functions and Then x()= where 0<fl<l, 0<0'<1. Therefore ^ Hence But 2 , where = Now let amp ( 1 + ^ j = r ; then, if cos(>/ ^)=^0, as w increases from to oo, 7r<V<7r. Hence, if n = OL+i/3, and if s>a J, / ie^t&V 1 --' \ 22 7 = 100] ASYMPTOTIC EXPANSIONS 273 Thus F^)!^, where M= ^"" -' Therefore It follows that 1 1 22 ;* But >; can always be chosen so that M is finite ; therefore, by sufficiently increasing \z\, | R, ] can be made arbitrarily small. Hence the series is asymptotic. Note 1. The expansion can be written (47i 2 -l 2 )(47i 2 -3 2 )(47i 2 -5 2 )(47i 2 -7 2 ) f (W-l 2 )(4? T" 2!(8^) '* '" 2. Since G_ n (z) = e inn G n (z), the expansion also holds whenR(n+J)>0. Asymptotic expansion ofj u (z). Again, since J n (ze iir ) = e i?lir J 7l (z)> we can write 7riJ n (z) = G n (z) - e i ' i ' r G n (ze i ' r ). Thus, if 7r/2<^0<C7r/2, the asymptotic expansion for J n (z) is given by __ " 2!(8z) 2 (4n-l s )(4w f -3 2 )(4n*-5 2 )(4n*-7 s ) 2 /(47i 2 -! 2 ) " / 2 /(47i 2 "V(7rz)l l! 8z 3!(8z) 3 M.F. 274 FUNCTIONS OF A COMPLEX VARIABLE [OH. xv Also, since 3 n (z) = e in "J n (ze- i7r ), it follows that, when the expansion is " 12 (4tt 2 -l)(4n*-3)(4n-6) 82 3!(8 2 js~ . Xsm COROLLARY. The difference between two consecutive zeros of J n (z) tends to the limit TT as z tends to infinity. Example. Prove where Jc is positive, and the quantities m g are the zeros of the function J (?w) regarded as a function of in. Since G (wa)J '(^)- J (na)G / (a)= , (cf. p. 241) 1 1 m s aJ l (m s a)' Now f5$ Jc FIG. 83. where C (Fig. 83) denotes a closed curve which crosses the .r-axis at the origin and at an infinitely distant point between two zeros of 3$(za\ and the summation extends to all positive values of m,. Therefore, since J (za) is an even function of 2, t J c But - Hence Lim =7r lf f J (ar)a-^flte. 100] EXAMPLES XV 275 EXAMPLES XV. 1. If K(/3)>0, R(y-j8)>0, shew that II p- l (l-W-*- l (l-ztr*dt-*(fr y-)F(a, ft y, 0). Use this formula to prove Gauss's Theorem. 2. If m is a positive integer, shew that deduce that 3. If wi is a positive integer, shew that [Use Ex. 1, 97.] 4. Use Ex. 1, 97, to prove that, if m and n are integers such that n ^ 0, m ^ - 7i, where C is a closed curve enclosing 5. Establish the formulae : (i) P^iW^^PT^+^ (iii) (7i- [Apply the method of partial integration to the definite integral form for P m (4] 6. Shew that the formulae of the previous example also hold for Q n w (z). 7. If | z | < 1, shew that i + 2\ r /l\ ) L \S) 2 according as 1(2)^0. [Use Exs. VIII. 20, and Ex. 2, 62.] 8. Prove that P n (^) = - tanmr{Q n (s)-Q_ n _i(z)K 276 FUNCTIONS OF A COMPLEX VARIABLE [OH. xv 9. Prove that [Use the second expression given in 97 for 10. Prove that [Use Exs. 9, 2, 97.] 11. Shew that (i) P M -( - z) = e^'^P n m (z) - - sin (n + m)7r- l Q B w (*), 7T according as 12. Shew that, if \z\>I, 2" +1 cosmr r 3 1\ ' 2' ?/ .on , , [Use Ex. 2, 97.] 13. Shew that, if |,|<1, 2 according as I (z} < 0. 14. Shew that 15. If R(w + *) > 0, shew that MISCELLANEOUS EXAMPLES. 1. Shew that ^ and give a geometrical interpretation of this equation. 2. If n is a positive integer, prove that (i) z 2n - a? n = (z 2 - a*) (z* - 2az cos - + a 2 . . . - 2az cos ( z 2 - 3. Prove that, if the points z lt z 2 , 2 3 , are the vertices of an equilateral triangle, *? + Z* + Z 3 2 = Zfa + V 3 + Zfr . 4. If 15 2 , z 3 , are the vertices of an isosceles triangle, right-angled at the vertex z 2 , prove that 5. If (! - 3 2 )(V - 2 ') = ( 2 - 3 )(V - 3 ') = (Z 3 - ZjXV - /), shew that the triangles whose vertices are 2 1} 2 2 > z zt an( ^ ^i'* V? ^3'? are lateral. 6. Similar triangles QRL, EPM, PQN, are described on the sides of the triangle PQR. Shew that the centroids of triangles PQR and LMN are coincident. 7. If !, 2 , 3 , and b l9 6 2 , 6 3 , are the vertices of two triangles which are directly similar, shew that any three points which divide the line joining the pairs of points a lt b ; 2 ^2 '> a ^ ^3 5 i n tne same ratio, form a third similar triangle. 8. If the lines joining z 2 and z 3 , z 3 and 2 15 z 1 and 2 , are divided in the same ratio r at z/, z 2 ', z 3 ', respectively, and if the triangles whose vertices are z u Z 2t z si an( i z ii z zi z s'i are similar, shew that either r= 1 or else both triangles are equilateral. 9. Let ABCD be a parallelogram of which AC is a diagonal, and let ABX, DCY, ACZ, be similar triangles. Prove that triangle XYZ is similar to each of them. 10. OCAD, OEBF, are circles, where O, A, B, C, D, E, F, are the points (0, 0), (2, 0), (6, 0), (1, 1), (1, -1), (3, 3), (3, -3), respectively. If w = *J{(I -z)(4-z)}, and if w=2 when z=0, find the values of w at A when z moves from O to A (i) along OCA, (ii) along ODA ; find also the values of w at B when z moves from O to B (i) along OEB, (ii) along OFB. Ans. -2*2 278 FUNCTIONS OF A COMPLEX VARIABLE 11. Shew that the equation w=\(z+^r^\ where z = re ie , determines a transformation which carries over circles, r= constant, and straight lines, 0= constant, into confocal ellipses and hyperbolas respectively. Sketch the system of confocals. If P is any point within the circle || = 1, shew that there is a point Q outside that circle which is carried over into the same point of the w-plane as P is transformed into. 12. If w=a(z c)/(z + c), where a and c are real and positive, shew that the interior of the circle | z \ = c in the z- plane corresponds to that half of the w-plane which lies to the left of the imaginary axis. 13. If w=l/z, and if the point z describes that part of the line 4 t y = 3(#- 2) which lies in the first quadrant, find the path described by the point w. Shew on the same diagram the path described by w when z describes that part of the line 4y + 3(.#-2) = which lies in the fourth quadrant. Indicate in each case the direction of motion. Ans. Those parts of the circles 6u 2 + 6v 2 =3u4:V which lie in the fourth and first quadrants respectively. 14. Shew that the transformation w=4/(z + l) 2 transforms the circle |^| = 1 into the parabola # 2 =4(1 u\ and that the interior of the circle corresponds to the exterior of the parabola. 15. Shew that all the roots of 2 5 +2,s 2 + 2 + 3 = are in absolute value less than 1-6. [Cf. the proof of the Theorem of 10.] 16. If a and b are real and positive, shew that the equation z* p + az + b=Q has 2p roots to the right, and 2p to the left, of the imaginary axis. If b is negative, shew that 2p+l roots lie to the right, and 2p-l to the left, of the imaginary axis. 17. If a and b are real, shew that the equation z* p - l + az + b=Q has 2jo or 2jp - 1 roots to the right of the y-axis, according as b is positive or negative. 18. Prove that : (i) Lim (sec z - tan z) = ; cos (ii) Lim -^. - = *; (iii) Lim ^ - -=f. ' z-^i sm-rrz z ^i simrz 19. Shew that sin 2# i si 20. If z tends to infinity along a straight line through the origin, shew that Lim tan 2= i, according as the line lies above or below the real axis. z >-oo 21. If w=coshzj shew that the whole w-plane corresponds to any strip of the 2-plane of breadth TT bounded by lines parallel to the #-axis. Also shew that, to the lines x= constant, y= constant, correspond the confocal ellipses and hyperbolas, %2 ^ _ ^2^2^ - 2 ~ ' cos 2 sin 2 ~" MISCELLANEOUS EXAMPLES 279 22. If w=\og{(z-a)/(z &)}, shew that the lines u= constant correspond to a coaxal system of circles whose limiting points are a and 6, whil6 the lines v= constant correspond to the orthogonal system. 23. If z=ctanh(7ri0), shew that the lines u=u correspond to the coaxal and the lines v = v Q to the orthogonal system of coaxal circles. 24. If the sequence z l1 z^ z 3 , ... , is convergent, shew that the sequence Zl+Z 2 Z 1 + 2 + Z 3 15 ~T~* ~~3 '-' converges to the same limit. 25. If the sequences z l , z 2t z 3 , ... , and z, z 2 ', 2 3 ', ... , converge to the limits z and / respectively, shew that the sequence i^, w 8 , w s , ... , where converges to the limit zz'. 26. Integrate e a2 /(l+ e 2 ), where 0<a<l, round the rectangle whose sides are x= E, y=0, y=2?r, and shew that r e ax dx IT 27. Prove that fe. |loglr,H-l<r<l, log ^ , if / < 1 or r > 1. Deduce that [log(cos 0) dO=j^ log (sin (9)rf(9=|logi. [Integrate -^A _| \ Z_ M_\ __J round the contour of Fig. 33, and put o?= tan #.] 28. If - 2 < a < 2, prove that Deduce that, if - 2 < a < 2, sin 20(tan ff^dB^j sin [Integrate a f 2 round the contour of Fig. 37.] 280 FUNCTIONS OF A COMPLEX VARIABLE 29. By integrating <>g and og * where r ftnd g ftre real r iz r+iz and positive, prove that 30. By integrating logf 1-f t*J ^ r 2 and logfl+i'-j 2 g .,, where r and are real and positive, prove that 31. If a, c, and m are real quantities such that m g 0, c> 0, shew that sinm(x-a) dx v /, - mc ~ ii p x-a 32. Shew that, if a and b are real, and m^tt^O, /" sinm(x a)siun(x b) j _ sinn(a b) -- _ - QjX - 7T - ; - -< x-a x b a-b 33. Prove that 34. If ^ r < 1, shew that tf<9 27T 'o l-Srcos^ + r 2 1-r 2 /*2ir ,7/3 35. Shew that J o y ^= 2?r or 0, according as | a \ < 1 or | a \ > 1 . 36. Shew that, if | a n \ ^ 1 for all values of n, the equation cannot have a root whose modulus is less than ^. Also shew that the only case in which it can have a root z=*e ie is when a n = e~* ne , (w = l, 2, 3, ...). 37. Shew that, if 38. If | z | < 1, shew that 39. Shew that ,.. T . ,_ . ,, > , x .. x T . 1 cos(l (i) Lim (l+cos7r2)/tan 2 7T2}=J ; (n) Lim 2 >0 MISCELLANEOUS EXAMPLES 281 40. Shew that, if | 2 | < 1 or | z ; > 1, the series _n ,. n 1 has the sum s/{(s- 1)(2 2 41. Prove that (i) | cos 2 1 ^ cosh 1 2 1 , (ii) \sinz |^sinh \z\. [Use the Taylor's Series for cos z and sin 2.] 42. If |s | >1, shew that z+I 2 2 +l 43. Shew that the series - -- 2! 3! is convergent if E(s)>0, divergent if E(s)<0. 44. Shew that the series is convergent for all values of z except 0, 1, - 2, - 3, ____ 45. Shew that, for points interior to the circle 3.r 2 + 3y 2 + 2.?; - 1 = 0, 1-* ' (l-zf (l-z) 3 ' "1-32 46. Prove that, if | z \ < 1, and the principal value of tan-^ is taken, 47. If \z\ <1, shew that 22 3*3 48. If . is neither zero nor a multiple of 2;r, shew that cosh 2- cos a. * f_ 2 2 ) 1 +: 49. Shew that sin* / 4_._.W, 4 . 2 \ /, 4 . 50. Shew that (i) 51. Shew that the series 1_J_ + 1 J , ~ 1 + 22 2 + 2 3~3+l represents a meromorphic function with simple poles at the points 1, -2, -3,.... 282 FUNCTIONS OF A COMPLEX VARIABLE 52. If a is positive, shew that f coscu; , Tre~ a f I (H^^TTi* 53. Shew that ,, ,.v sin7r(2 + c) z + c-^r, (, z \ - 54. Prove that (i) - . v = -- II. I 1- - }*i v ' SIRTTC c _ \ ?i-c/ ,... sin 2 ?!-.? " /, ^ 2 (ll) 1 -- r-s - = II 1 13 - " ( . 4z* ~\ sin 3z 55. Shew that II i 1 - 7 -oo I (?nr + zy) smz 56. Shew that ( " e- 2cose cos(^sin #)d<9 = |- f ^0 2i J o x W Shew that V X ^ sinh (77^2) + sin ( ' ~ co -r, ,, , " 1 58. Prove that 2 Si (?i+.r) 2 +3/ 2 y c 59. Calculate the residues of the function (l+s 2 )-"- 1 , and shew that dx 1.3.5....(2ra-l) 60. Shew that J x L = ^' 61. Shew that, if m ^0, a > 0, /oo ^j a / Q\ I ^ gciajasj ^-g-( m+'-}' 62. If - 1< E(0) < 3, shew that /" Jo o 63. If ft is a positive integer, shew that ose cos (^(9 - sin 64. If r\<l, shew that 65. Shew that I 7 Jc (z TTl ( where C denotes the circumference of the circle ^ 2 +y 2 -2^-2y=0 described positively. 66. If n is a positive integer, prove that P(cos fl-pgj {cos + j-^ .(- 2)9 MISCELLANEOUS EXAMPLES 283 [Expand both sides of the equation (i -sf cos tf+^-(i-f^a in powers of , and equate the coefficients of M .] 67. If n is zero or a positive integer, shew that (i) P,n +1 (o)=o, (ii) P 2 ^o)=(-ir 1 68. Shew that [Expand both sides of the equation and equate the coefficients of { 2n .J 69. OB is one diagonal of a square OABC which has the side OA on the .r-axis and the side OC on the y-axis ; through D(2a, 2a), the mid-point of OB, lines are drawn parallel to OA and OC so as to divide OABC into four equal squares with sides of length 2a. If w is given by the series 16 - 1 . 2m-lirx . 2n pr,ove that (i) w=0 along each side of the four squares ; (ii) w\ within each of the two squares about the diagonal ODB ; (iii) w= - 1 within each of the squares about the diagonal ADC. 70. Integrate (1 - e~*)/z round the contour consisting of the positive x and ?/-axes and a quadrant of an infinite circle, and shew that (i) 71. If b and r are positive, and a is real, prove that (ii) 72. Shew that, if o.>0, m>0, -Kr<l, x sin 2our TT -dx= f : sn GO? (io f-^ Jo m 2 + x*l- 2r cos 2out' + r 2 2 ( 1 + r) (e*" m - rV \ * / \ / flntegrate (i^ 284 FUNCTIONS OF A COMPLEX VARIABLE 73. Shew that the root of the equation z = +v:e z which has the value when 10 is given by n provided \iv\<\e~^~ l |. 74. If z=+esinz, shew that, for small values of e, /\ / e ! & 3 2 W ^=C+ r jSmf+2T e (ii) sin z = sin + ^ si sn cos + sn cos 75. If z = (+wz m+1 , where =0, and if that root of the equation is taken which has the value f when w=0, shew that provided | w | < | m w (wi + i)--if- | . 76. If n is a positive integer, shew that (i) P n '(^) = (2w (ii) P/(^) = (2?i 77. If n is a positive integer, shew that the n zeros of ~P n (z) are all real and lie between 1. [Apply Rolle's Theorem to (# 2 - l) n and its derivatives.] 78. Shew that, if n is zero or a positive integer, and if E(f)>0, T (cosh 2t-zfk'P n (z)dz 79. If | r | < 1, shew that n 9 cos 20 , cos 3-^ (i) r cos 9 - r 2 - + r 3 - -- . . . = i log (1 + 2r cos + r 2 ), A O r n _ ... where the principal value of the inverse tangent is taken. 80. Prove that, if < < TT, 81, Prove that (i) = -i log{4(cos 0-cosa) 2 }, (ii) cos cos a - i cos 2^ cos 2a. + J cos 3# cos 3a. - ... = Jlog{4(cos + cosa.) 2 }, unless one of the quantities 6 -a. and 6+a, is an even multiple of TT in case (i) or an odd multiple of TT in case (ii). MISCELLANEOUS EXAMPLES 285 82. Shew that, if ^ 6 ^ 2:r, (i) cos (9+^ cos2<9+^cos3(9+...= 1 3 2(3(9 2 - (ii) sm<9+j3sin2^ + ^3s 83. If ^ ^ TT, shew that -fsin 2 6log(4sin 2 0), , x sin 40 sin 66 sin 80 . n/1 , _ m . 9/3 (n) Y^+-JT^ + ~3~T + ''' = * m ^~ ^ - sin cos 6 log (4 sin 2 0). 84. If n is a positive integer, and if 1 2 1 < 1, shew that Deduce that _ 85. If ^ 6> ^ TT, shew that TT ,. x sin 6 sin 3^ sin 86. If - 7T/2 ^ ^ ^ 7T/2, shew that (9 2 \ . sin ZQ sin 5(9 87. If - 7T/2 ^ ^ ^ 7T/2, shew that cos 30 cos 50 cos 76 TT oo a 'I2 88. Shew that the series =i J represents a continuous function in the part of the 2>plane for which 100 = 0, and that the function is holomorphic at all points below the real axis. Ort -r, 1 2 / COS27T^ COS37TJI' \ 89. Prove that ;y = - + -^( COSTTJ; -- ,^5 + ^ -- ... o 7T \ A" v J represents a series of equal and similar parabolic arcs standing in contact along the a'-axis. 90. Prove that J\ - - .* 70X = 21 coth ira coth 7r6. m^-oo ,,= C/ 2 91. If -KR(a)<l, shew that rsinh ajc dx -- cosh.r a? 286 FUNCTIONS OF A COMPLEX VARIABLE Deduce that, if A. is real, rsin Xx dx . / , vrA\ r -- = 2 tan- 1 1 tanh I cosh# x \ 4 / 92. Prove that Lim ( 1 + \ + \ + . . . +^- T - \ log M) = log|2 + |y. ,t_> C) o \ o o zn \.& / 93. Shew that 94. Shew that [Use the identity ( -!)-- 2(0"* - 1 )-' = ( 2 + 1)- 1 .] [Shew that e- a - ; e- 2a +... + (- 1 ) n ~ 1 6- na = ^. f ^^ z e ~ azdz > where C is tlie contour of Fig. 58, and use Ex. 94] 96. Shewthat (i) cot=l-B 1 -B 2 -B 3 -... , 2!4 4!6 6! 97. Prove that (i) l-^^-i, (ii) w =2 l\ */ J H^Afr+l/ 3 98. If U(n)> 0, shew that (cos ^) n ~ 1 cos (a tan ^) cos (w + 1) dO = P(cos 1 ^) n - 1 sin (a tan <9)si ^o [Use Exs. VIII. 6.] 99. Shewthat *() 100. Shew that, if m is a positive integer, 101. Shew that (i) V^)= 2 , (ii) ^(0) = 2 , (iii) V^(-i) = - 102. Prove, by using the equation MISCELLANEOUS EXAMPLES 287 that, if R(c)> - 1, 103. Shew that, if B(0)> - 1 and R(fc)>l, e -tz t n~\ ) H (0 + 2)" (0 + 3) n r(w)Jo e'- 104. If E(a)>0, shew that 105. Shew that, if R(0)> - 1, 106. Shew that, if R (71) > 1 , cos < n = ~ cos ;<) cos If 7i is zero or a positive integer, prove, by considering the cases n even and n odd separately, that Jo c -~v-T-/x"~-r/ -r ..(n + k\(n + k n \ (n+k ^M(-i- l )~\-^- n Deduce that, if n is zero or a positive integer, ^ l"^^./7f\/ / \* 71 W I 7T Jo rTTi ^57Ti V^cfo, where z Q = ae 4 , =ae 4 , and the path of integra- tion is a semi-circle of centre the origin and radius a described positively. Also find the values of the integrals which have Z Q as initial point, and whose paths are : (i) a complete circumference of the circle ; (ii) two cir- cumferences ; (iii) three circumferences. What is the shortest non-zero path from ZQ along the circumference which makes the integral zero ? 4 / ^''X A / ? -'\ Ans. -fV&a*; (i) -|a*(l+?J, (ii) -fHe 3 +e 3 ), (iii) 0; three- fourths of a circumference. 108. Prove that f ;/RT _|L___ =F (,, |). 109. Shew that all elliptic integrals /E(^, JX.)dx, where R(A', y) is rational in x and ?/, and X is a cubic in x with no repeated factors, can be expressed in terms of integrals of the three types f dy f ydy f dy J VC--)' J ^--- J -' 288 FUNCTIONS OF A COMPLEX VARIABLE 110. Establish the identity where the product is taken for all integral values of A and /x from to n t with the restriction 111. Shew that W (n+l)* 112. Shew that 113. Shew that where pM 114. Prove that du 115. The function <(p(u} has a real period 2^ and an imaginary period 2o> 2 , where w 2 --= ^log ( - ), a and 6 being real and positive, and such that a<b. TT \aj Shew that, if 0=fj(f^llogi ), the annulus in the ^-plane bounded by the \ 7T Oil circles |f| = a and |f| = 6 and a barrier along the positive real axis, corre- sponds to the entire 2-plane. Shew also that only one point of the annulus corresponds to each point of the 2-plane. 116. Prove that sn 117. Shew that / \ - sn(- ,) = Lim -*o M* 8 118. If six of the nine points in which the cubic y 2 = 4x 3 -g 2 x-g 3 is cut by a second cubic lie on a conic, shew that the other three points lie on a straight line. 119. If a conic passes through four fixed points on the cubic y*=^-g&-gv shew that the straight line joining the two variable points of intersection passes through a fixed point on the cubic. MISCELLANEOUS EXAMPLES 289 120. Solve the equation w" + az 2 iv = Q. 121. Solve the equation vf" + -Izw' + w = 0. Ans. w i = l -^ + -^T zQ - ' 9'; 1 3 , 3.9 3.9.15 2 10 +..., 1 2_A*5 5.11 8 5.11.17 n W *~2\ Z ~5~! 8! Z 11! 122. If n is a positive integer, shew that all the zeros of P n (z) are simple zeros. [By differentiating Legendre's Equation it can be shewn that if P(.z) has a zero of the second order, ~ P n (z) = for all positive integral values of .] 123. Find ttiat integral of the equation which has the value unity when z = 0. Ans. Find regular integrals in the neighbourhood of 2 = for the equations of Examples 124-128. 124. 42%" + 4zw'- Ans. 2 1\ 23 /2 2 1 ^ =wz+z 'l-^r 125. oo -n+4 Ans. wz^e* w 126. z*(z + l)w"-z*w' + %(Zz+l)w = 0. Ans. w^z, w 2 127. 22 2 (2-2)w // -2(4-2)w' + (3-0)w = 0. Ans. w 1= z^ w 2 =(z- 128. 2 2 (l-2)^y" + 2(5^-4)w' + (6-92)^<;=0. J?i5. w^^ 3 , w> 2 = ^ log + z 2 . 129. If n is zero or a positive integer, shew that dz n+ 130. Shew that, for all values of M.F. 290 FUNCTIONS OF A COMPLEX VARIABLE 131. Shew that, for all values of n, (i) (1 -z^r n (z}=nP n ^(z)-nzP n (z), (ii) (l-^)PLl (iii) (1 -z*)Q n (z (iv) (1 -^QL.M + nQnW -nzq n 132. If n is zero or a positive integer, shew that where W n -i(z) is a polynomial of degree n l. [in Ex. 1, 90, write <*,=! ^ df-| ^ ? 133. With the notation of Example 132, shew that [Substitute the expression obtained for Q n () in Ex. 132 in Legendre's Equation, put W w _ 1 (2) = a 1 P n _ 1 (^)-f-a 3 P n _ 3 (^)+... , and use Example 76.] 134. Shew that deduce that, if m and ^ are positive integers, both odd or both even, while if m is an even and n an odd integer, r p m (z)p w (s)^=(-i) j -1 ,\ 2 27V 2~\ [Cf. proof of Exs. XIV. 5, and use Ex. 131.] 135. If m and n are integers such that w^O, m^w, shew that Pn~ m O?)= ^gn^; 136. Prove that, if E(n) > R(m) > - 1, and deduce the results : 9 ^n (i) Jn(z) = ^ ~ M2 ) M ^ COSigMrfM > where (ii) _ = J (^sin 6)sin 2 JO [Expand J TO (2w) in powers of w, and integrate.] MISCELLANEOUS EXAMPLES 291 137. Solve the equation zi&" + (n + l)u/-w = 0. An*. w^s-Sj^V:), 138. Shew that, if n is an odd positive integer, [Use the formula 2.J n (z) = J n _i(.r) + J n+ i(4] 139. Tf is an integer, shew that [In EXS. XIV., 14, put f=e ifl ,and = <j>-irl2.] 140. Prove that ,? 2 = 2 (2rc) 2 J 2 (4 n=l [Differentiate the equation e^ : ^- 1 /0= 2Jt*){" w ^h regard to ^, multiply by , differentiate again, and put = 1 .] 141 . Prove that (i) cos x = J (x) - 2 J 2 (a?) + 2 J 4 (.r) - . . . ; (ii) sin x = 2.1, (.?;) - 2 J 3 (.r) + 2 J, (.r) [In Exs. XIV., 14, put =i.] 142. Shew that (!) J) 143. Shew that 144. Shew that, if p is a positive integer, [Use Exs. XIV., 11.] M.F. T2 292 FUNCTIONS OF A COMPLEX VARIABLE 145. If n is a positive integer, shew that ' (iii) r 146. Prove that T MJ M= [Shew that the coefficient of f - j in the product i and apply Gauss's Theorem.] J47 Shew that, if n is zero or a positive integer, 2 r* - / J n (2z cos 0) cos (M) o?</) = J ll+t M J )( _ A . (:). 7T JO Y~ ~2~ [Expand J n (2iCos<^) in powers of cos<, and use Examples 106 and 146.] 148. If x and u are real, prove that [Use the relation - "!'' '- = uS i prove that 149. If .r is real, and [Use Exs. XIV., II.] 150. If and 6 are positive constants, prove that .r cos / * 30 / [Put J O (&F) = ~ | cos (6. of integration.] 151. If Ii(b ?)>0, shew that Jo ^ [Put J (a.r) = - I cos (a.r cos (Exs. XIV., 18), and change the order and change the order of integration; or, expand J Q (CLT) in powers of .T, and integrate term by term (cf. Bromwich, Infinite Serf ex, 176, B).] MISCELLANEOUS EXAMPLES 293 152. Shew that, if R(2?i + 1)>0, and E(6 ia)>0, [For (i) use the substitution given in Exs. XIV., 18, for J n (#.r), and change the order of integration ; after the first integration expand (6 + ia cos c^)- 2 "- 1 in powers of cos <, and integrate again ; or, expand J(o#) in powers of #, and integrate term by term. For (ii), differentiate (i) with regard to b.] 153. Prove that - / * e** cos * cos (y sin <) d<j> = J 7TJO [Expand cos (y sin (/>) in powers of sin 0, and apply 99, Cor., Example 145, (iii), and Taylor's Theorem.] INDEX. The numbers refer to the pages. Abel's test for convergence of series, 80. theorem on continuity of series, 125. Addition of complex numbers, 1, 3. Amplitude, 2, 4. of a function, variation of, 11-17. principal value of, 2, 4. Argand diagram, 2. Argument, 2. Asymptotic expansions, 136. (See under Bessel, Euler, Gamma function.) Bernoulli numbers, 132. Bessel function, 236. Bessel function G n (z), 240. addition theorem for, 254. as a contour integral, 267. asymptotic expansion of, 271. in terms of Bessel functions of first and second kinds, 240. recurrence formulae for, 241. Bessel function of first kind, 237. addition theorem for, 254. as a contour integral, 267, 268, 270. as a function of its order, 239. asymptotic expansions of, 273, 274. recurrence formulae for, 239. zeros of, 241, 274. Bessel function of second kind, 238, 239. Bessel functions, relations between, 241. Bessel's equation, 236, 266. Beta function, 144, 145. Binomial theorem, 90. Branch of function, 13. Branch point, 14, 39. of an integral, 257. Cauchy's integral theorem, 51, 54. residue theorem, 57. Circular functions, 33, 83, 90. Coefficients, undetermined, 9G. Collinearity of points on cubic, 197. Complex numbers, 1. geometrical representation of, 1 . operations with, 1-5. Complex variable, 7. function of a, 7. path of variation of a, 7. Conformal representation, 37. Congruent points, 179, 180. Conjugate numbers, 1, 2. Connected region, 30. Continuation, analytical, 122, 208. of hypergeometric function, 153, 156, 249. of integral of diif. equation, 213. theorems on, 123, 124. Continuity, 23, 24. Abel's theorem on, 125. of series, 92. uniform, 24-26. Convergence of infinite product, 107, 108. unconditional, 107. Convergence of sequence, 42. uniform, 42. Convergence of series, 76. absolute, 76, 78. circle of, 80. of a double series, 78. of power series. 80, 82, 95. radius of, 80. ratio tests for, 77. uniform, 92. Coordinates, polar, 2, 29. rectangular, 2. Cross-cut, 30. Derivative, 26, 28. of function of a function, 30. of holomorphic function, 28, 70. of inverse function, 30. partial, 31, 70. INDEX 295 The numbers refer to the pages. Determinant of fundamental system, 216, 223. index of, 224. Differential equation, homogeneous linear, 209. coefficients of, 210. construction of, 216. domain of ordinary point of, 210. dominant equation, 210. Frobenius' method of solution, 225. fundamental equation, 220. fundamental system, 215, 257. indicial equation, 225. integrals of, 210. of the first order, 210. of the second order, 210. ordinary point of, 210. singularity of, 210. solutions of, 210. Differentiation, 26, 28, 29. of series, 93. under integral sign, 44, 69, 138. Discontinuity, removable, 23. Division of complex numbers, 1, 4. Domain of a point, 38, 210. Elements of a function, 208. Elliptic function, 180. order of, 181, 182. poles of, 180 to 183. zeros of, 182. (See under Jacobian and Weier- strassian functions.) Elliptic integrals, 169. reduction of, 170-173. transformation of, 170-174. (See also Legendre's and Weier- strass's elliptic integrals.) Equations, roots of, 16, 69. Euler's constant, 135. asymptotic expansion of, 134. Euler's definition of gamma function, 141. Expansion, Lagrange's, 119. Expansion of functions in scries of fractions, 103, 105. Exponential function, 32, 90. Fourier series, 86. Frobenius' method of solving linear diff. equations, 225. indicial equation, 225. solutions free from logarithms, 228. uniform convergence of series with regard to index, 227. Fuchsian type, equations of, 243. sum of indices a constant, 243, 244. Function, analytic, 29, 208. conjugate, 31. continuous, 23. dominant, 210. doubly-periodic, 179. elements of a, 208. even, 33, 97. geometrical representation of a, 7, 10. holomorphic, 29, 52, 93. initial value of, 10. integral, 88. integrals of, 48. inverse, 30. limit of, 22. meromorphic, 39, 40, 89, 160. multiform or multiple-valued, 7, 161, 209. odd, 33, 97. of a complex variable, 7. of a function, 24, 30, 49. of two complex variables, 69, 137. of two real variables, 26. periodic, 32, 86. periodic, of the second kind, 187. periodic, of the third kind, 189. rational, 89. rational integral, 88. region of existence of, 7. regular, 29. simply-periodic, 86. single-valued, 7, 209. transcendental integral, 88. uniform, 7, 209. uniform, classification of, 88. Fundamental equation, 220. Fundamental system of integrals, 215, 257. associated with fundamental equa- tion, 220. in neighbourhood of singularity, 219. Fundamental theorem of algebra, 68, 69. Gamma function, 109, 139, 141. asymptotic expansion of, 146. duplication formula for, 145. Euler's definition of, 141. expression as a contour integral, 143. Gauss's definition of, 141. the derived function \f(2), 141. Gauss's differential equation, 228, 258. function 11(2), 141. sum, 117. theorem, 144. 206 FUNCTIONS OF A COMPLEX VARIABLE The numbers refer to the pages. Geometrical representation. (See under Complex numbers, Functions and Transformations. ) Green's theorem, 45. Gregory's series, 84. Harmonic functions, 31. Harmonics, cylindrical, 236. spherical, etc., 249. Hyperbolic functions, 33, 90. Hypergeometric equation, 228, 258. relations between integrals of, 249. the twenty-four integrals of, 247. Hypergeometric function, 77, 151, 229, 246, 247. analytical continuation of, 153, 156, 249. as a contour integral, 259. Hypergeometric series, 77, 78, 144. convergence of, 77. Identities, 83. Image of point, 9. Indented contour, 65. Indicial equation, 225. fundamental system, 227. Infinity, point at, 9. continuity at, 23. integral at, 51, 137, 139. integrals of diff. equ. at, 212, 213, 224. loop about, 162, 168. residue at, 58, 96. singularity at, 38, 39. Integrals, contour, 59, 97, 113. convergent, 136. curvilinear, 42. definite, 48. double, 69, 138. elliptic, 169. evaluation of definite, 59, 97, 113. finite moduli of definite, 50. Fresnel, 62. indefinite, 53. independent of paths, 52. limiting values of definite, 60, 63, 113, 115. of holomorphic functions, 50-52. of meromorphic functions, 160. of multiform functions, 161. piincipal values of, 65. uniformly convergent, 137. with infinite paths, 51, 137, 139. Integrals of differential equation, 210. analytical continuation of, 213. at infinity, 212, 213. Integrals of differential equation, existence of, 210. fundamental system of, 215. in form of infinite series, 213. initial values of, 210. linearly independent, 215. Integrals of diff. equ. in form of definite integrals, 255, 266. branch points of, 257. fundamental system of, 257. Integrals of diff. equ. near a singularit} 7 , 219. at infinity, 224. fundamental system of, 219. index of, 222. regular, 222. Integrand, infinite, 136, 139. Integration, change of order of, 69, 138. of series, 93. partial, 53. under integral sign, 69, 138. Invariants (see under Weierstrass). Inverse points, 9. Inverse sine function, 163. Inverse tangent function, 34, 84, 86. I(p) notation, 1. Jacobian elliptic functions, 167,182, 198. addition theorems for, 202. complementary modulus of, 166, 167, 200. derivatives of, 200. diff. equ. of quarter periods of, 176, 231. duplication formulae for, 204. Legendre's relation for, 175. moduli of, 167, 200. orders of, 182, 202. periods of, 167, 201, 202. poles of, 200, 202. relations between, 200. relations between periods of, 202. relation to Weierstrassian functions, 201. residues at poles of, 202, 205. transition from Weierstrassian func- tion to, 198. zeros of, 200. Jacobi's imaginary transformation, 205. Lagrange's expansion, 119, 125. . Landen's transformation, 174. Laplace's equation, 31. Laurent's series, 84, 95. absolute convergence of, 85. INDEX 297 The numbers refer to the pages. Legendre functions, 249. of the first kind, 214, 235, 236. of the second kind, 235, 236. recurrence formulae for, 236, 252, 289, 290. Legendre polynomials, 99, 214, 235. expression in series, 102, 103, 121. in definite integral forms, 100-102. integrals involving, 122. recurrence formulae for, 102, 124, 129. Rodrigues' formula for, 120. Legendre's associated equation, 249, 259. Legendre's associated functions, 250. as definite integrals, 260-263. relations between, 251, 262-265, 275, 276. Legendre's complete elliptic integrals of the first and second kinds, 174. Legendre's equation, 213, 234. relation to Gauss's equation, 235. Legendre's first normal elliptic integral, 163, 173. inversion of, 166, 201. Legendre's normal integrals, 173. Legendre's relation, 175, 188. Limit, 22. at infinity, 22. infinite, 23. of a sequence, 42. of function, geometrical illustration, 22. of ratio of two functions, 30, 83. uniform convergency to a, 23. Liouville's theorem, 68. Logarithmic function, 34-36, 83, 161. Logarithmic transformation, 35. Loops, 145, 162, 164, 168. about point at infinity, 162, 168. notation for negative, 162. Mittag-Leffler's theorem, 105. Modulus, of complex number, 2, 3, 4. (See under Jacobian elliptic func- tions.) Multiplication of complex numbers, 1, 4. Naperian logarithms, 34. Numbers, complex, imaginary, real, 1,2. geometrical representation of, 1-6. Orthogonal systems, 32. Path of variation, 7, 10, 22. Period, of a function, 80, 179. parallelogram, 179. primitive, 86, 179. P-function, Biemann's, 244. Point at infinity, 9, 38, 39. Points, congruent, 179, 180. Points, critical, 38. Points of inflection on cubic, 197. Points, ordinary, 38, 210. Points, singular, 38. Pole, 38, 39, 67, 118. an isolated singularity, 39. at infinity, 38, 88. of order n, 38, 86. principal part at a, 86. simple, 38. Polynomials, 88. Power, the generalised, 36. Product, infinite, 107, 108. expression of function as, 108, 109. Quantities e, rj, positive, 23. Region, closed, 92. connected, 30. function holomorphic in, 53. multiply-connected, 30, 47, 58. of existence of function, 7. of uniform convergence, 92, 96. simply-connected, 30. Residue at a pole, 57, 58, 67, 96. at infinity, 58, 96. Riemann's P-function, 244. indices of, 245. in terms of hypergeometric functions, 246. Rodrigues formula, 120. Root extraction, 1, 5, 36. Roots of equations, 4, 5, 10, 69. theorems on, 118, 119. R(p) notation, 1. Sequence, 42. Series, convergent, 76. multiplication of, 77, 82. power, 80, 82, 95, 125. uniformly convergent, 92. Sigma functions, 109. duplication formula for, 190. elliptic functions in terms of, 190. properties of, 189. Similar figures, 8, 37. Singularities, 38. at infinity, 38, 39, 88, 89, 106, 181. 298 FUNCTIONS OF A COMPLEX VARIABLE The numbers Singularities, essential, 39, 86, 89, 90, 106, 181. isolated, 38, 39. line of, 101. non-essential, 39. of a diff. equ., 210. Stirling's formula, 150. Sturm's theorem, 16. Subtraction of complex numbers, 1, 3. Summation of series by residues 116. Summation of trigonometrical series 126, 127. Tangent to a cubic, 197. Taylor's series, 82. 95. absolute convergence of, 83. Transformations, 7. bilinear, 8, 9. geometrical representation of, 8. linear, 7, 8. rational, 8. (See under Landen, Logarithmic.) Trigonometrical series, summation of, 126. Uniformly convergent series, 92, 127. continuity of, 92. differentiation of, 93. integration of, 93. power series, 95. Weierstrass's M test for, 94. , Variable, complex, 7. independent, 7. Vectors, 2. refer to the pages. Weierstrassian elliptic function, 106, 169, 180. addition of semi-period to argument, 187. addition theorem, 185. diff. equation satisfied by, 183. duplication formula for, 186. elliptic functions in terms of, 191. geometric application of, 196. in terms of sigma functions, 190. invariants of, 184, 194. Legendre's relation for, 188. order of, 182. periods of, 169, 180, 195, 196. poles of, 181, 182. relation to Jacobian functions, 201. residue at pole of, 181. transition to Jacobian functions, 198. values when one period real and one purely imaginary, 194. zeros of first derivative of, 182, 184. Weierstrassian elliptic integral, 167, 185, 195, 196. inversion of, 169, 185. Weierstrass's theorem, 108. Weierstrass. (See under Sigma and Zeta functions and Uniformly con- vergent series.) w-plane, 10. Zeros, 1,39,67, 118, 119. of order n, 39, 83. simple, 39. Zeta functions, Weierstrass's, 106. elliptic functions in terms of, 188. properties of, 187. z-plane, 2, 10. GLASGOW : PRINTED AT THK UNIVERSITY PRESS BY ROBERT MACLETIOSE AND TO. LTD. WORKS FOR STUDENTS OF HIGHER MATHEMATICS An Introduction to the Theory of Infinite Series. By T. J. I'A BROMWICH, M.A., F.R.S. 8vo. 155. net. Introduction to the Theory of Fourier's Series and In- tegrals and the Mathematical Theory of the Conduction of Heat. By Prof. H. S. CARSLAW, M.A., Sc.D., D.Sc. 8vo. 145. net. Elliptic Functions. By A. C. DIXON, M.A. Globe 8vo. 55. A Treatise on Differential Equations. By A. R. FOR- SYTH, F.R.S. 8vo. 145. net. Treatise on Bessel Functions. By Prof. A. GRAY and Prof. G. B. MATHEWS. 8vo. 145. net. Differential and Integral Calculus. By Sir A. G. GREENHILL, M.A., F.R.S. Third Edition. Crown 8vo. los. 6d. Applications of Elliptic Functions. By Sir A. G. GREENHILL, M.A., F.R.S. 8vo. 123. Manual of Quaternions. By Prof. C. J. JOLY, M.A. 8vo. i os. net. Introduction to Quaternions. By P. KELLAND, M.A., and P. G. TAIT, M.A. Revised by Dr. C. G. KNOTT, D.Sc. Crown 8vo. 75. 6d. The Theory of Determinants in the Historical Order of Development. By Sir T. MuiR, C.M.G., M.A., LL.D., F.R.S. 8vo. Vol. I. 175. net. Vol. II. The Period 1841 to 1860. 175. net. The Theory of Relativity. By L. SILBERSTEIN, Ph.D. 8vo. i os. net. Short Course in the Theory of Determinants. By Prof. L. G. WELD, M.A. Crown 8vo. 8s. net. LONDON: MACMILLAN & CO., LTD. WORKS FOR STUDENTS OF PHYSICS Vectorial Mechanics. By L. SILBERSTEIN, Ph.D. 8vo. 7s. 6d. net. Studies in Radioactivity. By Prof. W. H. BRAGG, F.R.S. 8vo. 55. net. Studies in Terrestrial Magnetism. By Dr. C. CHREE, F.R.S. 8vo. 55. net. Researches in Magneto-Optics. By Prof. P. ZEEMAN, Sc.D. 8vo. 6s. net. Electric Waves. By H. HERTZ. Translated by D. E. JONES, B.Sc. 8vo. IDS. net. Miscellaneous Papers. By H. HERTZ. Translated by D. E. JONES, B.Sc., and G. A. SCHOTT, B.Sc. 8vo. ros. net. Modern Theory of Physical Phenomena, Radio-activity, etc. By AUGUSTO RIGHI. Translated by A. TROWBRIDGE. Crown 8vo. 55. net. Electromagnetic Theory of Light. By C. E. CURRY, Ph.D. Part I. 8vo. 125. net. Electric Waves. By Prof. WILLIAM S. FRANKLIN. 8vo. I2s. 6d. net. Application of Dynamics to Physics and Chemistry. By Sir J. J. THOMSON, O.M., F.R.S. Crown 8vo. 75. 6d. Utility of Quaternions in Physics. By ALEX. M'AuLAY. 8vo. 55. net. "^ The Mathematical Theory of Perfectly Elastic Solids, with a Short Account of Viscous Fluids. By W. J. IBBETSON. 8vo. 2is. The First Three Sections of Newton's Principia. With Notes, Illustrations, and Problems. By P. FROST, M.A., D.Sc. 8vo. I2S. LONDON: MACMILLAN CO., LTD. INDINGLIST 23 1951 ,0 ec ft S3 O H Cj Q EH CO 0> +3 P o OfiH University of Toronto Library DO NOT REMOVE THE CARD FROM THIS POCKET Acme Library Card Pocket LOWE-MARTIN CO. LIMITED